Kinematics

This second section covers key areas such as motion and its type, equations of motion and graphical representation of motion, Newton’s three laws of motion, Pressure and Pascal’s principle. The first point of discussion will be on motion.

This introduces you to various types of motion with respect to the position of an object and how these happen in real-life situations. You will further be led through a series of explorative activities that will guide you to understand the three laws formulated by Sir Isaac Newton, the basic concepts and principles behind each law, and how different variables interconnect to describe the motion of objects. Newton’s laws of motion talk about Newton’s three laws of motion, which explain the relationship between the motion of a physical object and the forces acting upon it. By applying Newton’s laws to real-life situations, you can predict the motion of objects experiencing forces and anticipate outcomes in various scenarios. Pressure and Pascal’s principle will be discussed as well. The lessons on Pressure and Pascal’s principle, which are fundamental concepts in fluid mechanics and the basis for brake systems and hydraulic press, provided you with essential knowledge about the concept of pressure, its relationship with depth and the principles of hydrostatics. You will understand that pressure in a fluid increases with depth. There are practical demonstrations and experiments related to the brake system and hydraulic press to help foster a hands-on application of all these Engineering concepts. You will further appreciate the significance of pressure transmission in enclosed fluids, realising how a small force applied at one point can result in a significant force at another. Moreover, an understanding of pressure informs various scientific and industrial applications, contributing to advancements in technology and everyday practices.

At the end of this section, you should be able to;

• Describe the various types of motion, i.e. circular, oscillatory, rectilinear, spin and random.

• Establish equations of uniformly accelerated motion and its application in daily life.

• Represent the motion of objects graphically, i.e. distance-time, displacement-time and velocity-time and deductions that can be made from it.

• State Newton’s laws of motion.

• Identify daily applications of Newton’s laws of motion.

• Apply Newton’s second law to establish the relationship between force, mass and acceleration.

• Explain how pressure changes with depth in a fluid.

• Explain the operation of brake systems in vehicles and the operation of the hydraulic press.

• State Pascal’s principle.

Key Ideas

• Motion is explored through key aspects such as circular motion (motion along a circular path), oscillatory motion (repetitive back-and-forth motion), rectilinear motion (straight-line motion), spin motion (rotation around an axis), and random motion (unpredictable motion). It also involves the equations of uniformly accelerated motion and interpreting distance-time, displacement-time, and velocity-time graphs.

• Newton’s laws of motion include three principles: The first law, the second law and the third law of motion.

• Newton’s First law of motion: An object remains at rest or in uniform motion unless acted upon by a resultant force.

• Newton’s Second law: The resultant force on an object equals its mass times its acceleration (F=ma).

• Newton’s Third law: Every action has an equal and opposite reaction.

• Newton’s laws explain everyday phenomena such as car acceleration, seatbelt use, and gun recoil. Applying Newton’s second law helps establish the relationship between force, mass, and acceleration, showing that force is directly proportional to acceleration and inversely proportional to mass.

• Pressure in a fluid increases with depth due to the weight of the fluid above. This principle is utilised in hydraulic systems such as vehicle brake systems and hydraulic presses, which use incompressible fluids to transmit force through confined spaces.

• Pascal’s principle states that a change in pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid and to the walls of its container.

Understanding these concepts is crucial for various fields such as physics and engineering, providing the foundation to analyse and predict the behaviour of moving objects in both academic and practical contexts.

Types Of Motion

Motion is the change in position of an object over time relative to a reference point. It is described by parameters such as displacement, distance, velocity, acceleration, time, and speed.

This lesson takes you through activities that will help you understand how things move differently in the world. Through your explorations, you will unlock the secrets of how objects travel. This is important to help you understand how things behave in nature and in the technology, we use every day.

Begin your journey of exploration.

There are different types of motion you can explore.

Activity 2.1 Exploring the types of motion

1. Work individually or in pairs.

2. Research on any one of the following types of motion (you may choose one for yourself, or your teacher may decide for you):

a. Rectilinear

b. Circular

c. Oscillatory

d. Rotational

e. Random (Brownian)

3. Record your research findings along the following guidelines:

a. What are the characteristics of the motion?

b. What are some examples of the motion?

c. What are some real-life applications of the motion?

Use diagrams, graphs, animations and real-life examples to illustrate your findings.

4. Present your findings to other students who have investigated a different type of motion from yours.

5. For each type of motion presented to the class, complete the table below by providing a detailed description/definition, drawing a diagram, and giving a relevant example. Analyse the types of motion and relate them to real-world applications.

An example is given for rectilinear motion, but you are welcome to add further detail following the presentations.

Note: Definitions and examples can be found in Annex 2.1.

Table 2.1: Summary of types of motion, diagrams and real-world application Type of Motion Description/ Definition Diagram

Example & Real-World Application Rectilinear Motion Motion along a straight line.

Example: A car driving on a straight highway.

Application: Used in

analysing motion in transportation systems.

Circular Motion

Rotational/ Spin Motion

Oscillatory Motion

Random Motion

Activity 2.2 Exploring types of motion through models Research the internet to explore how the types of motion have been explained using models.

You may also explore some models in your Physics laboratory or the science section of a library with the help of your teacher.

Note that conclusions can be found in Annex 2.3.

A. Rectilinear Motion Model

Objective: Explore characteristics of linear motion using a toy car and a ramp.

Materials needed:

• Toy car or small wooden cart

• Wooden plank or cardboard (for ramp)

• Stopwatch or timer

• Measuring tape or ruler What to do:

1. Set up the ramp.

2. Release the car and measure time and distance to calculate average speed (= distance/time).

3. Experiment with different inclined planes (adjust the height of the ramp by hand or by resting the end on a pile of books).

4. Plot a graph of average speed against the angle between the ramp and the

table.

5. Discuss the impact of ramp angle on average speed.

6. Predict the effects of changing ramp surface.

B. Circular Motion Model

Objective: Explore circular motion using a ball or weight tied to a string and passed through a hollow tube, with more weights hung from the bottom of the string (see diagram below).

Materials needed:

• String or twine

• Small ball or stone

• Hollow tube (e.g. pen casing)

• Paper clip

• Hanging masses

• Ruler

• Stopwatch What to do:

1. Set up the equipment as shown in the diagram below.

2. Swing the ball in a circular path, trying to keep the paperclip in a constant position.

3. Calculate the frequency of the rotation by using the formula frequency = 1 / time period.

4. Discuss forces keeping the ball in motion.

5. Analyse the effects of changing the weight of the hanging masses on the frequency of the orbit required to keep the paperclip in a constant position.

C. Oscillatory Motion Model

Objective: Explore oscillatory motion using a pendulum.

Materials needed:

• String or twine

• Small rock or seed

• Stopwatch or timer What to do:

1. Swing the pendulum and measure oscillation time.

2. Experiment with different lengths and weights.

3. Discuss the impact of length and weight on the time for one complete swing (time to return to original position).

D. Rotational Motion Model

Objective: Explore rotational motion using a spinning object.

Materials needed: Small flat object (coin or bottle cap) What to do:

1. Spin the object and observe rotation.

2. Note duration and speed changes.

3. Discuss factors affecting spin duration.

Equations of Motion

Equations of motion provide a framework for understanding and predicting how objects move, and their applications are applied to various aspects of our daily lives such as transportation, sports, technology, etc.

Let us do an activity to help us define the different types of motion.

Activity 2.3 Definitions of types of motion

1. With the person sitting next to you, or in groups, come up with your own definitions for the following terms:

a. Displacement

b. Initial velocity

c. Final velocity

d. Acceleration

e. Time

f. Average velocity

g. Instantaneous velocity

2. Deriving the Equations of Motion

Use the graph below, showing the motion of an object accelerating uniformly from a velocity u to a velocity v over a time t, to derive the equations of motion.

Fig. 2.1: A velocity-time graph for deriving equations on motion You may or may not choose to do this with the guidance of your teacher.

The gradient of the graph gives the acceleration, a, of the object.

Gradient = change in y________ change in x

a. Give an expression for this in terms of a, v, u and t.

b. Rearrange to make v the subject to arrive at the first equation of motion: v = ……….

The area under the line gives the displacement, s, of the object:

Area of trapezium = area of rectangle + area of triangle

a. Give an expression for this in terms of s, v, u and t

b. Replace (v-u) with (at), as per the first equation of motion This is the second equation of motion, s = …….

Make t the subject of the first equation, and substitute this into the second equation. Make v²the subject.

This is the third equation of motion, v²= ……… Extension Question: Can you also get to equation 3 via a different method, using the formula for the area of a trapezium to find the displacement?

Now let us apply these concepts to solve some questions.

Activity 2.4 Application of the equations of linear motion

Worked Example 2.1

A car accelerates from rest at a constant rate of 3 m/s². How long will it take for the car to travel a distance of 150 meters?

Solution

Identify the known quantities and desired quantities:

Initial velocity, u = 0 m/s (since the car starts from rest) Acceleration,a = 3 m/s²Distance, s = 150 m The desired quantity is time.

Choose the appropriate equation of motion:

The equation that relates distance, initial velocity, acceleration, and time is:

s = ut + 1__ 2at²Substitute the known values into the equation:

150 = 0 × t + 1/2 × 3 × t²Simplify the equation:

150 – 3/2 × t²Solve for t:

Multiply both sides by 2:

300 = 3t²Divide both sides by 3: 100 = t²t = √___ 100 t = 10s Answer: The car will take 10 seconds to travel 150 meters.

Trial Questions 1

1. A car travelling at 20 m/s begins to decelerate at a constant rate of 2 m/s².

How far will the car travel before coming to a complete stop? Additionally, how long will it take for the car to come to rest?

2. A cyclist accelerates from 5 m/s to 15 m/s over a distance of 50 meters.

What is the cyclist’s acceleration?

3. A car travelling at 30 m/s sees an obstacle 100 meters ahead and decelerates uniformly at 5 m/s². Determine whether the car will stop before hitting the obstacle. If it stops, calculate the distance from the obstacle where the car stops. If it does not stop, calculate the speed of the car when it reaches the obstacle.

Motion Under Gravity

Motion under gravity refers to the movement of an object that is influenced by the force of gravity. This force causes objects to accelerate downward toward the Earth at a constant rate, denoted by g, which is approximately 9.8 ms⁻².

Conventionally we treat downwards as being the positive direction. Hence,

1. When the body is moving downwards, the acceleration (g) due to gravity is positive (i.e. +g) hence the equations become, v = u + gt …………………(4) h = ut + 1__ 2gt²………………(5) v²= u²+ 2gh ………………(6)

2. When the body is moving or is thrown upwards, the acceleration due to gravity (g) becomes negative (i.e. -g) hence the equations become:

v = u – gt …………………(7) h = ut – 1__ 2gt²………………(8) v²= u²– 2gh ………………(9) Where;

v = final velocity g = acceleration due to gravity h = displacement (vertical distance fallen) u = initial velocity (usually 0 ms⁻¹as the object starts from rest) g = acceleration due to gravity (9.8ms⁻²on Earth) t = time elapsed Trial Questions 2

1. A stone is dropped from the top of a cliff that is 80 meters high. Calculate

(a) the time it takes for the stone to hit the ground and (b) the velocity with which it hits the ground. Assume there is no air resistance.

2. A rocket is launched vertically with an initial velocity of 50 m/s and accelerates uniformly at 4 m/s² for 12 seconds. After this period, the rocket’s engine stops, and it continues to move upwards under the influence of gravity (assuming g =10 m/s²) until it reaches its maximum height. Determine the maximum height attained by the rocket.

Graphical Representation of Motion of Objects

Graphs are often used to represent the motion of objects over time. The most common graphs used for motion are:

• Distance-time

• Displacement- time

• Velocity-time Let us explore these three types of graphical presentations.

Distance-Time Graphs

Worked Example 2.2

Scenario 1 A car travels at a constant speed of 60 km/h. use the data provided below to plot a distance-time graph and calculate the total distance travelled by the car Distance (km) 0 60 120 180 240 300 360 Time (hr) 0 1 2 3 4 5 6

Solution

Since the car is travelling at a constant speed, the distance increases linearly with time. For every hour, the car travels 60 kilometres.

• After 1 hour, the distance is 60 km.

• After 2 hours, the distance is 120 km.

• After 3 hours, the distance is 180 km. etc The graph of this scenario will be a straight line with a slope representing the speed of 60 km/h.

Fig. 2.2: A distance-time graph Total distance travelled:

• The car travels for 6 hours.

• Distance = Speed × Time = 60 km/h × 6 h = 360 km.

Trial Questions 3

1. Amina spends her day running errands and visiting a friend. She walks from her house to the park in 5 minutes, covering 150 meters steadily. She rests at the park for 5 minutes, keeping her distance from home constant. Then, she walks 100 meters to the grocery store, reaching it by the 15th minute.

After shopping for 5 minutes with her distance constant at 250 meters, she walks 300 meters to her friend’s house, arriving by the 30th minute. She stays there for 10 minutes, keeping her distance constant at 550 meters.

Finally, Amina walks back home, covering the 550 meters in 10 minutes and returning by the 50th minute. Based on this scenario, create a table of the data and create a distance-time graph to represent Amina’s journey.

2. The table below shows the time and distance for a runner to complete a race.

Distance (meters) 0 50 100 150 200 Time (seconds) 0 10 20 30 40 Use this data to:

a. draw the distance-time graph representing the runner’s motion.

b. calculate the total distance of the runner.

Share your work with a friend or subject teacher.

Position-Time graphs Position can be positive or negative depending on the reference point (‘in front’ of the reference is positive and ‘behind’ of the reference is negative).

Trial Questions 4

1. How far does Kwaku walk from his home, and in what time frame, if he starts at 0 metres, walks 300 metres to the park in 10 minutes at a constant speed, rests at the park for 5 minutes with his position unchanged, and then walks back home at the same constant speed in 10 minutes? Also calculate his velocity at each section of his journey.

2. Akua goes for a morning walk from her house to the nearby market and then returns home.

Use the table provided, draw a displacement-time graph representing Akua’s walk.

Position (metres) 0 100 200 200 0 Time (minutes) 0 5 10 15 20 Calculate Akua’s average speed during her walk. Assume Akua walked for 20 minutes and covered a total distance of 400 meters. Give your answer in m/ minute, m/hr and m/s.

Velocity-Time Graphs

The key features of a velocity-time graph are as follows:

1. Slope: The slope of the graph indicates acceleration. A positive slope indicates positive acceleration, a negative slope indicates deceleration, and a zero slope indicates constant velocity.

2. Area under the curve: The area under the graph represents displacement (the total distance travelled).

3. Zero Velocity: When the line touches the horizontal axis, the object is at rest.

Worked Example 2.3

A car accelerates from rest to 20 ms⁻¹in 5 seconds, travels at constant velocity for 10 seconds, then decelerates to a stop in 5 seconds. This data is given in the table below:

Time (seconds) 0 5 10 15 20 Velocity (m/s) 0 20 20 20 0

i. On a sheet of graph paper, draw a horizontal line and label it (Time(s)) and scale it from 0 to 20 seconds with 5-second intervals.

ii. Draw a vertical line intersecting the horizontal-axis at 0. Label this line velocity (ms⁻¹) and scaled it from 0 to 20 ms⁻¹in 5 ms⁻¹intervals.

iii. Plot the points.

Fig. 2.3: Velocity-time graph

iv. Use a ruler to draw straight lines connecting the points in sequence.

Calculating the total displacement (total area under the velocity-time graph) Look at the velocity-time graph and identify the geometric shapes formed under the graph.

• From 0 to 5 seconds: A triangle (acceleration phase)

• From 5 to 15 seconds: A rectangle (constant velocity phase)

• From 15 to 20 seconds: Another triangle (deceleration phase)

1. Calculate the Areas:

a. Triangle ABF (0 to 5 seconds):

Base = 5 seconds Height = 20 m/s Area = 1/2 × Base × Height Area = 1/2 × 5 × 20 = 50 m

b. Rectangle BCEF (5 to 15 seconds):

Base = 10 seconds Height = 20 ms⁻¹Area = Base × Height Area = 10 × 20 = 200 m

c. Triangle CED (15 to 20 seconds):

Base = 5 seconds Height = 20 ms⁻¹Area = 1/2 × Base × Height Area = 1/2 × 5 × 20 = 50 m

2. Sum the Areas:

Total Area (displacement) = 509 m + 200 m + 50 m = 300 m Alternatively, the formula for calculating the area of trapezium could be used Area = 1/2(a + b) × h Where:

a and b are the lengths of the two parallel sides (also called bases).

h is the height (perpendicular distance) between the two parallel sides.

From the graph Area = 1/2 (20 + 10) × 20 Area = 300 m Trial Questions 5

1. A runner starts from rest, accelerates to 8 m/s in 4 seconds, maintains this speed for 6 seconds, and then decelerates to rest in 4 seconds. Plot the velocity-time graph and use the graph to calculate the total distance.

2. Draw a motion graph and an accompanying story that includes various types of motion, including periods at which the object is stationary, accelerating, decelerating, and moving at a constant speed.

3. Analyse the information shown in the distance-time graph below to instead plot a velocity-time graph showing the same information. Compare with a peer or teacher.

Extension task: No solutions Motion Practical and Graph Skills Equipment and materials:

• a ramp resting on a wooden block (or a pile of books, etc)

• a trolley or toy car

• a metre rule

• a set square or protractor

• a second wooden block (to stop the trolley)

• a stopwatch to time the trolley Use the equipment listed above to measure instantaneous velocity against time and plot a graph of your results to find the acceleration of the trolley. Instantaneous velocity can be found by marking the track at 5-10cm intervals and using the formula: speed = distance/time for each interval. A slow-motion camera may be helpful.

Newton’s Laws of Motion

Welcome. Today, we are going to explore the exciting world of motion through Newton’s laws. These three fundamental principles explain how everything around us moves, from the smallest particles to the largest planets. First, we will learn about Newton’s three laws of motion, which describe how objects behave at rest, in motion, and when forces interact. Next, we will look at how these laws apply to our everyday lives (think about how cars accelerate, how we walk, and even how sports work). Finally, we will dive into Newton’s second law to understand the relationship between force, mass, and acceleration. Get ready for some fun

activities and real-life examples that will make these concepts come alive. Let us get started on this journey of discovery and see how Newton’s laws shape the world around us.

Let us dive into the fascinating world of Newton’s laws of motion, formulated by Isaac Newton to describe how objects move when forces act on them. These laws are fundamental in physics and explain so many things we observe every day.

Activity 2.5

1. Draw diagrams to represent the direction, magnitude and nature of the forces acting on objects in the following situation:

A vehicle has broken down, and as responsible citizens, you know that it needs to be towed off the road to prevent other vehicles from crashing into it.

2. Compare with a peer or teacher.

Newton’s First Law: The Law of Inertia

A body will continue in its state of rest or uniform motion in a straight line unless acted upon by an unbalanced force.

Inertia describes the tendency or reluctance of an object to alter its state of motion.

This tendency means that an object at rest will resist being set in motion, while an object in motion will resist any attempts to change its velocity or direction.

Think about this: Imagine you are in a moving vehicle, and it suddenly stops. Your body tends to keep moving forward because of inertia, and you might feel a jerk. Have you ever felt that? Why do you think seatbelts are important? Share your thoughts with a friend.

Activity 2.6 Demonstrating inertia with a pile of books and a tablecloth Materials needed:

• a smooth table

• a tablecloth (preferably smooth and thin)

• a pile of books (start with a few lightweight books and gradually increase the weight) What to do:

1. Preparation:

a. Lay the tablecloth flat on the table, ensuring there are no wrinkles or folds.

b. Place the pile of books in the centre of the tablecloth. Make sure the books are stacked neatly and stably.

c. What do you think will happen to the books when we pull the tablecloth? Why?

2. Share your thoughts with a friend.

3. Investigate:

• Stand at one end of the tablecloth and firmly grip the edges.

• Quickly and smoothly pull the tablecloth horizontally. Aim to pull the tablecloth out from under the books without disturbing their position.

• Observe the books as the tablecloth is pulled away.

4. Share your observations with a friend.

Consider:

a. Why did the books stay in place when you pulled the tablecloth quickly?

b. How does this show Newton’s First Law of Motion?

c. What happened when you pulled the tablecloth slowly? Why was it different?

d. Share your observations with a friend or subject teacher

Activity 2.7 Understanding Newton’s first law (law of inertia) Materials needed:

• toy car or any small, wheeled object

• smooth surface (table, floor)

• small objects (books, blocks) to create an inclined plane

• notebook and pen for recording observations Scenario: Imagine a vehicle has broken down and needs to be moved off the road to prevent other cars from crashing into it. Discuss with a family member or friend what makes the vehicle move or stay still.

What to do:

Experiment:

1. Place the toy car on the smooth surface.

2. Give the car a gentle push and observe how it moves and eventually stops. Discuss with your partner what causes it to stop.

3. Create a small, inclined plane using books or blocks and cardboard or plank.

4. Place the toy car at the top of the inclined plane and observe that it stays still until you give it a push. Discuss why it does not move until a force is applied (gravity acting along the incline).

Consider:

a. Why does the car stop moving on the flat surface?

b. What external force acts on the car to stop it?

c. How does the inclined plane affect the car’s motion?

Newton’s Second Law: The Law of Acceleration

The rate of change of momentum of a body is directly proportional to the resultant force applied, and it happens in the direction of the unbalanced force.

Activity 2.8 Understanding Newton’s Second Law

Materials needed:

• Same toy car or a small, wheeled object

• Weights (coins, small stones)

• Smooth surface

• Ruler and stopwatch (optional)

• Notebook and pen for recording observations Scenario: Continuing with the broken-down vehicle scenario, consider how hard you would need to push it if it were heavier. Discuss with your partner how mass and force are related to moving the vehicle.

What to do:

1. Experiment: Place the toy car on the smooth surface and push it with a consistent force.

2. Observe how fast it moves and discuss with your partner.

3. Add some weights (like books) to the car.

4. Push the car again with the same force. Observe the difference in speed and acceleration compared to the unweighted car.

5. If available, use a stopwatch to time how long it takes for the car to travel a certain distance with and without weights.

6. Record your observations and calculate the force using F = ma if you have the measurements.

Consider:

a. How did adding weights affect the car’s speed?

b. How would you describe the relationship between mass, force, and acceleration?

c. What would happen if you pushed a heavier car with the same force as a lighter car?

Activity 2.9 Understanding Newton’s second law (law of acceleration) Materials needed:

• A toy car (to represent the broken-down vehicle)

• A smooth surface (like a table or floor)

• A piece of string

• Weights (like small stones or coins)

• A stopwatch (optional) What to do:

1. Setup:

a. Attach a piece of string to the front of the toy car.

b. Tie weight to the other end of the string to act as the force pulling the car.

c. What do you think will happen when you push the toy car gently?

d. What if you pull it with different weights?

2. Explore:

a. Add a small weight to the string and observe the acceleration of the toy car.

b. Gradually add more weight and observe how the acceleration changes.

c. Use a stopwatch to measure the time it takes for the car to travel a certain distance with different weights if desired.

d. How did the acceleration of the toy car change when you added more weight?

e. What will happen to the acceleration if you add more weight to the car? Will it increase or decrease? Explain.

Newton’s Third Law: The Law of Action and Reaction For every action, there is an equal and opposite reaction.

For example, just as we are pulled towards the centre of the Earth with a force equal to our weight, the centre of the Earth is pulled towards us with an equal and opposite force.

Another example is that the force of air resistance acts on a parachute as it descends through the atmosphere, colliding with particles in the air. Equally, there is a force acting on the particles that is equal and opposite.

Activity 2.10 Demonstrating Newton’s third law Materials needed:

• Sheets of paper (for making boats)

• A large bowl or container filled with water

• Small sticks or straws (for making paddles)

• Tape or glue (for securing paddles) What to do:

1. Build the Paper Boats:

a. Follow simple origami instructions to fold sheets of paper into small boats.

b. Ensure the boats are stable and can float on water.

2. Prepare the Bowl: Fill a large bowl or container with water, deep enough to float the paper boats.

3. Make the Paddles:

a. Use small sticks or straws as paddles.

b. If needed, tape or glue small pieces of paper or cardboard to the ends of the sticks to act as paddles.

4. Perform the Demonstration:

a. Place the paper boats in water.

b. Use the paddles to push water backward, mimicking the action of rowing.

c. Observe the motion of the boats as you push the water.

d. Explain your observations.

Activity 2.11 Demonstrating Newton’s Third law of Motion Materials needed:

• Balloon

• Straw

• String (long enough to stretch across a room)

• Tape

• Two fixed points (like chairs or door handles) to tie the string What to do:

1. Setup:

a. Stretch the string across the room and secure it tightly between two fixed points, like chairs or door handles.

b. Thread the straw onto the string before tying the second end. Make sure the straw can slide easily along the string.

2. Prepare the balloon:

a. Inflate the balloon but do not tie it. Pinch the end to prevent the air from escaping.

b. Tape the inflated balloon to the straw, positioning it so that the open end faces one of the fixed points.

c. Hold the balloon at one end of the string.

d. Release the balloon and observe its movement along the string.

e. Watch as the balloon moves in the opposite direction to the escaping air.

f. Share your observations with a friend about how the air pushing out of the balloon (action) causes the balloon to move in the opposite direction (reaction).

3. Consider:

a. Write down or discuss how this experiment illustrates Newton’s Third Law.

b. Think of other examples where action and reaction forces are observed, such as in rocket launches or swimming.

Activity 2.12 Exploring the effect of force in different scenarios.

Perform investigations on each scenario below, and then discuss the discussion questions with your peers or teacher.

Scenario 1: When the object is already in motion and experiences a resultant force Materials needed:

• toy car or any small, wheeled object

• smooth surface (table, floor)

• small weights (coins, small stones)

• ruler and stopwatch (optional)

• notebook and pen for recording observations What to do:

1. Place the toy car on a smooth surface.

2. Give the car a gentle push to set it in motion.

3. While the car is moving, apply an additional force by pushing it again or stopping it with your hand.

4. Observe how the car’s motion changes (e.g., it speeds up, slows down, or stops).

5. Record your observations.

Discussion questions:

i. What happens to the car’s speed when you push it harder?

ii. What happens when you try to stop it?

Scenario 2: When the object is already in motion and experiences no resultant force.

Explore:

1. Place the toy car on a smooth surface.

2. Give the car a gentle push to set it in motion.

3. Let the car move without applying any additional forces.

4. Observe how the car continues to move and eventually stops.

5. Record your observations.

Discussion questions:

i. How does the car’s speed change over time?

ii. What causes the car to stop eventually?

Scenario 3: When the object is stationary and experiences a resultant force Explore:

1. Place the toy car on a smooth surface and let it sit still.

2. Apply a force by giving the car a gentle push.

3. Observe how the car starts moving from rest.

4. Record your observations.

Discussion questions:

i. What happens to the car when you push it from a stationary position?

ii. How does the car’s motion change immediately after the push?

Scenario 4: When the object is stationary and experiences no resultant force.

Explore:

1. Place the toy car on a smooth surface and let it sit still.

2. Do not apply any force to the car.

3. Observe how the car remains stationary.

4. Record your observations.

Discussion questions:

i. What happens to the car when no force is applied?

ii. Why does the car remain stationary?

Reflection

1. Summarise what you learned about how forces affect the motion of an object in different scenarios.

2. Explain how your observations relate to the concepts of resultant forces and motion.

3. Share your experiences with a friend or class teacher.

Relationship Between Force, Mass and Acceleration

Using Newton’s Second Law

An unbalanced force occurs when the forces acting on an object do not cancel each other out, resulting in a net force that causes the object to accelerate in the direction of the net force. This means the sum of all forces on the object is not zero.

Trial Questions 6

Use the hints below to complete the following force diagrams:

• Forces in the same direction: Add their magnitudes.

• Forces in opposite directions: Subtract their magnitudes.

1. + = 4N 7N

2. + = 15N 15N + = 15N 15N

3. + = 50N 23N Trial Questions 7 Draw the force diagrams in the following scenarios. Compare your answers with a peer or the teacher.

Example:

A Tug-of-war where team A is stronger than team B:

Scenario 1: A ball is lying on the ground.

Forces to consider:

a. The weight of the ball (acting downward)

b. The normal force from the ground (acting upward) Scenario 2: A person is pulling a box across the floor with a rope.

Forces to consider:

a. Tension force in the rope (acting horizontally, towards the person)

b. Frictional force opposing the motion (acting horizontally, opposite to the direction of motion)

c. Weight of the box (acting vertically downward)

d. Normal force from the ground (acting vertically upward) Scenario 3: A pendulum is swinging back and forth. Draw a couple of different versions of your diagram with the pendulum at different positions in its swing.

Forces to consider:

a. Tension force in the string (acting along the string)

b. Weight of the pendulum bob (acting downward) Newton’s Second Law (law of acceleration): The rate of change of momentum is directly proportional to the unbalanced force applied and it takes place in the direction of that force.

Δp/t ∝ F Or F = ma

Activity 2.13

Materials needed:

• Small toy car

• Smooth surface (like a table or floor)

• Newton meter

• Stopwatch

• Measuring tape or ruler

• Small weights (such as coins or small bags of sand)

• String What to do:

1. Explore:

a. Attach a piece of string to the front of the toy car.

b. Mark a starting line and a finish line 1 metre apart on a smooth surface.

c. Pull the car with a gentle, consistent force (use a Newton meter or hang a mass off the other end of the string and allow it to freefall over the edge of the table) over the 1 metre distance and measure the time it takes using the stopwatch.

d. Record the time and calculate the average velocity (= distance / time). Double your result to get a rough estimate of the final velocity.

e. Calculate the change in momentum (Δp ⃗ = mΔv ⃗ ).

f. Add small weights to the car to increase its mass and repeat the process.

g. Observe how the change in momentum varies with different masses.

h. Complete the table below.

2. Table to fill out:

Trial Mass

(kg) Time (s) Initial Velocity

(u) (m/s) Final Velocity (v)

(m/s) Change in

Velocity (Δv) (m/s) Change in

Momentum (Δp) (kg·m/s) Rate of Change

in Momentum Δp___ Δt (kg·m/s²) 1 0 2 0 3 0 4 0 5 0 Discussion questions:

i. What did you observe about the car’s motion when you pulled it with a gentle, consistent force?

ii. How did the car’s speed (velocity) change when you added more mass to it?

iii. What happened to the time it took for the car to travel 1 metre when you increased its mass?

iv. What do your observations tell you about the relationship between force, mass, and acceleration?

Review Questions 2.1

1. A stone is thrown vertically upwards with an initial velocity of 20m/s.

Determine the time it takes for the stone to reach its maximum height.

2. A stone is launched vertically upwards from the ground with an initial velocity of 50 ms⁻¹.

Determine:

a. The maximum height reached by the stone.

b. The time it takes for the stone to reach its maximum height.

c. The total time the stone is in the air before hitting the ground again.

3. Kwesi drives his taxi from Kumasi to Tamale, making a stop for fuel halfway through the journey. He spends 20 minutes refuelling before continuing the trip back to Kumasi.

a. Using the provided table, draw a detailed position-time graph illustrating Kwesi’s taxi ride. The graph should include labelled points for significant events such as the start of the trip, the fuel stop, arrival at Tamale, departure from Tamale, and return to Kumasi.

Table 2.2 Positions and time for Kwesi’s taxi ride Position(km) 0 240 240 0 Time(minutes) 0 120 140 160

b. Calculate Kwesi’s instantaneous speed when he is halfway to Tamale and when he is refuelling. Use the position-time graph to determine these speeds. Give your answers in km/hr.

c. Determine the total distance Kwesi travelled during his entire trip.

4. A car starts from rest, over the first 5 seconds, it accelerates at a constant rate, reaching a velocity of 20 m/s. For the next 10 seconds, the car cruises at this constant velocity of 20 m/s. Suddenly, the driver notices a red light and brakes, causing the car to decelerate at a constant rate over 5 seconds, reducing its velocity to 0 m/s. The car then stops at the traffic light, with its velocity remaining zero.

a. Sketch the velocity time graph of the motion.

b. Calculate the total distance travelled by the car.

5. A cyclist accelerates from 0 to 5 m/s in 2 seconds, travels at 5 m/s for 3 seconds, decelerates to 2 m/s over 2 seconds, travels at 2 m/s for 2 seconds, and then comes to a stop over 1 second. Plot a velocity-time graph and use it to find the total distance travelled.

6. A bus starts from rest and accelerates uniformly at 2 m s⁻²for 10s. It maintains the maximum speed attained for further 10 s and decelerates at 1 m s⁻²gradually to rest.

a. Draw the velocity-time graph.

b. Use the velocity-time graph to determine:

i. the maximum velocity attained.

ii. the time taken for the bus to decelerate to rest.

iii. the total distance covered

iv. the average velocity

7. Design Challenge Question: Design and build a model of a roller coaster that demonstrates and applies linear equations of motion. Your roller coaster model should simulate the motion of a car along its track, emphasising concepts such as acceleration, velocity, and distance travelled.

Use mathematical equations to predict and analyse the car’s motion at different points along the track. Consider incorporating loops, hills, or other features to showcase the principles of uniform motion and constant acceleration.

Present your design to your peers or teachers.

Review Questions 2.2

1. A concrete slab of mass 32 kg is being pulled along the ground with a force of 21 N. If the opposing frictional force is 5 N, calculate the acceleration.

2. A man of mass 2500 g is standing on a weighing scale in a lift. If the lift moves upwards with an acceleration of 3 m s⁻²and then downwards with the same acceleration, find the weight on the weighing scale when

a. He’s accelerating upwards

b. He’s accelerating downwards

c. the lift moves with a constant velocity of 6 m s⁻¹upwards

d. when lift moves with a constant velocity of 5 m s⁻¹downwards

3. A rocket of mass 2000 kg experiences a thrust force of 50000 N upward and a gravitational force of 19600 N downward. What is the net force and acceleration of the rocket?

4. Design an experiment to demonstrate Newton’s 1st Law of Motion using a ball, a flat, smooth surface, a starting line and a spring launcher. Describe the setup, procedure, expected results, and how these results support Newton’s 1st Law.

5. Analyse the forces acting on a car accelerating from rest to 60 km/h in 10 seconds. Determine the net force required if the car’s mass is 1500 kg, considering both the ideal scenario and the impact of friction and air resistance of 500 N.

6. Evaluate the propulsion mechanism of a rocket in space using Newton’s 3rd Law. Explain how this law applies to the thrust generated and the motion of the rocket and compare this with a real-life example of a balloon releasing air.

7. Describe a real-life scenario where Newton’s 1st Law applies to a moving object.

8. Compare the behaviour of a soccer ball on grass versus on a smooth gym floor in terms of Newton’s 1st Law.

9. Explain how seat belts in cars illustrate Newton’s 1st Law.

10. If a rocket with a mass of 2000 kg needs to achieve an acceleration of 10 m/s², what amount of thrust is required?

11. Analyse how increasing the mass of an object affects the force needed to achieve the same acceleration.

12. Discuss how friction affects the net force acting on an object and its resulting acceleration.

13. Analyse the forces involved when two ice skaters push off from each other.

14. Design Challenge Question: Design and create a homemade project that demonstrates and applies all three of Newton’s laws of motion. Your project should clearly show how Newton’s laws govern the motion of objects. Consider using everyday materials to build a model. Be prepared to explain the scientific principles behind your project and how it applies Newton’s laws.

Review Questions 2.3

1. A scuba diver, Kwesi is descending to a depth of 30 meters in the ocean.

The density of seawater is 1025 kg/m³ and the acceleration due to gravity is 9.8 m/s².

a. Calculate the pressure at a depth of 30 meters.

b. Explain how the pressure experienced by Kwesi changes as they descend deeper.

2. A hydraulic lift is used to raise a 2000 kg car. The piston area of the lift is 0.05 m².

a. Calculate the pressure required to lift the car.

b. Explain how applying pressure in a hydraulic system allows for the amplification of force.

3. A water tower has a height of 50 meters and a tank diameter of 10 meters.

The density of water is 1000 kg/m³.

a. Calculate the pressure at the base of the water tower.

b. Explain how the water tower design takes advantage of the relationship between pressure and depth

4. Otu is swimming at a depth of 10 meters in a freshwater lake. Calculate the pressure experienced by the Otu at this depth, given that the density of freshwater is 1000 kg/m³.

5. Yakubu lifts a box that weighs 200 N. The bottom of the box has a surface area of 0.4 m². What is the pressure the box exerts on the surface it is resting on?

6. Adwoa weighs 800 N and stands on two feet, each with a surface area of 0.02 m². What is the pressure exerted by the person on the ground?

7. A weightlifter, Bashiru lifts a 50 N weight on a 0.2 m²surface. What is the pressure?

8. Design Challenge Question: Design and build a model hydraulic arm that demonstrates the practical application of Pascal’s principle. Your model should include components illustrating how pressure transmitted through a confined fluid can amplify force and achieve mechanical advantage. Use everyday materials to construct a functional prototype capable of lifting a small load or performing a task, emphasising the principles of fluid pressure and hydraulic systems.