Dynamics

In previous lessons, you have seen that moving objects have motion described by speed and velocity. But in order to analyse the behaviour of a moving object, it is often not only the speed that matters—how heavy the object is also makes a difference. For example, it is much easier to stop a rolling football than to stop a moving wheelbarrow. This idea combines both the mass of the object and its velocity, and together they form a quantity called momentum. Momentum therefore helps to explain why heavier or faster objects are harder to stop.

Figure 9.6: Kicking a football at rest Think about kicking a football. When the ball is at rest, it has no momentum. The instant it is kicked, the force from the foot is applied for a short time, making the ball gain speed. This effect of a force acting over time is called impulse. The harder and longer the foot pushes on the ball, the greater the impulse, and the faster the ball moves away. Impulse is the change in momentum of the object.

Linear Momentum

The momentum of a body is the product of mass and velocity which is the measure of the body’s motion. It is a vector quantity has the unit of kilogram-metres per second (kgm/s).

Momentum = mass × velocity ρ = mv For an object speeding up or slowing down Final momentum = mass × final velocity (mv) Initial momentum = mass × initial velocity (mu) Change in momentum = final momentum – initial momentum (mv – mu) Newton’s second law of motion (Revision from Year 1) Newton’s second law of motion states that the time rate of change of momentum is directly proportional to the force applied and it takes place in the direction of the force.

Mathematically, F ∝ mv − mu/t F = k mv − mu/t But k = 1 F = mv − mu/t Impulse Impulse is the effect of a force acting on an object for a period of time. Impulse can be calculated as the product of a force acting on a body and the time during which it acts.

Mathematically, I = Ft From Newton’s second law F = mv − mu/t Therefore, I = Ft = mv − muThis is the Impulse-Momentum theorem which states that the impulse acting on an object is equal to the change in its momentum.

Examples of Impulse in Everyday Life

1. Catching a Ball

A cricketer moving their hands back while catching a ball increases the time of impact. This reduces the force on their hands, making it easier and safer to catch the ball.

2. Airbags in Cars

Airbags deploy during a collision to increase the time it takes for a passenger to come to a stop. This reduces the force of impact on the passenger, minimising injuries.

3. Jumping onto a Soft Surface When someone jumps from a height onto a soft surface like sand, their body sinks into the sand, increasing the time it takes to stop. This reduces the force of impact on their legs and feet, preventing injury.

4. Bending your Knees When Landing Similarly, bending your knees when landing from a jump increases the time it takes to stop, reducing the impact force on your legs.

5. Hitting a Baseball or Golf Ball

A good swing in baseball or golf involves a follow-through, which increases the time the bat or club is in contact with the ball. This imparts a larger impulse, resulting in a faster and farther-travelling ball.

6. Car Crashes

In a car crash, the forces involved act over a short period, causing a rapid change in momentum. Features like airbags and seatbelts are designed to mitigate the effects of these impulsive forces.

7. Martial Arts

Martial artists use the concept of impulse to generate powerful strikes. The rapid, forceful movement of their limbs creates a large impulse, transferring a lot of momentum to their target.

Conservation of Momentum

The law of conservation of momentum states that for an isolated system (one not acted upon by external forces), the total momentum of the system remains constant. That is the total momentum before collision equals the total momentum after collision. This means that momentum is not created or destroyed but rather transferred between objects within the system.

Isolated (or ‘Closed’) System A system where the net external force is zero. While internal forces (forces between objects within the system) can change the momentum of individual objects, they do not change the total momentum of the entire system (i.e. the momentum lost by one object would be gained by the other(s)).

For a system of two interacting objects (e.g., m₁and m₂):

Total initial momentum = total final momentum pᵢₙᵢₜᵢₐₗ = p_(final) m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂ Real Life Examples of Conservation of Momentum

1. Billiards/Pool When a cue ball strikes another ball, the total momentum of the system (both balls) before and after the collision remains the same. The momentum lost by the cue ball is gained by the other ball(s).

2. Car Accidents

In a car collision, the total momentum of the vehicles before the crash is equal to the total momentum after the crash, provided no external forces like braking or skidding are dominant. Accident reconstruction experts use this principle to analyse accidents and determine speeds.

3. Firing a Gun

When a gun is fired, the forward momentum of the bullet is balanced by the backward momentum (recoil) of the gun. The total momentum of the system (gun and bullet) remains zero, as it was before firing.

4. Rockets Rockets are propelled forward by expelling hot gases backward. The momentum gained by the rocket is equal and opposite to the momentum of the expelled gases, demonstrating conservation of momentum.

5. Jumping from a Boat Pushing off a dock to get onto a boat causes the boat to move backward in the opposite direction, illustrating the transfer of momentum.

6. Ice Skaters

When ice skaters push off each other, they move in opposite directions with equal and opposite momenta.

7. Jellyfish and Squid Propulsion

Jellyfish and squid use a similar method to propel themselves through water.

They expel water from their bodies, gaining momentum in the opposite direction.

Verification of Newton’s Third Law from Momentum Change Newton’s third law of motion (Review from Year 1) Newton’s third law of motion states that to every action there is an equal and opposite reaction.

Action = − Reaction Consider two bodies 1 and 2 of masses m₁ and m₂ acting mutually on each other.

Let their initial velocities be u₁ and u₂ respectively and let their final velocities be v₁ and v₂ respectively.

From the Impulse-Momentum theorem For body 1: I₁ = F₁ t = m₁ v₁ − m₁ u₁ For body 2: I₂ = F₂ t = m₂ v₂ − m₂ u₂ From the principle of conservation of momentum m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂ m₁ u₁ − m₁ v₁ = m₂ v₂ − m₂ u₂ −( m₁ v₁ − m₁ u₁) = m₂ v₂ − m₂ u₂ − I₁ = I₂ − F₁ t = F₂ t − F₁ = F₂ This derivation shows that if momentum is conserved in an isolated system, then the forces between interacting objects must be equal in magnitude and opposite in direction, which is precisely Newton’s Third Law. It underscores the fundamental nature of momentum conservation as a consequence of how forces interact.

Collision In physics, a collision is any event where two or more objects exert forces on each other for a short period. These interactions can involve direct contact, like a ball hitting a bat, or influence at a distance, such as two charged particles repelling each other. Momentum is always conserved in a collision, meaning the total momentum of the system remains constant. Kinetic energy, however, may or may not be conserved, leading to different types of collisions.

Types of collision Elastic Collision An elastic collision is a collision in which both momentum and kinetic energy are conserved. This means the total kinetic energy of the system before the collision is equal to the total kinetic energy after the collision.

Figure 9.7:A diagram showing elastic collision Characteristics

1. Total momentum is conserved.

2. Total kinetic energy is conserved.

3. No deformation of objects or generation of heat/sound due to the collision itself (in an ideal elastic collision).

4. Objects usually rebound off each other without sticking.

Mathematical conditions Conservation of momentum m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂ Conservation of kinetic energy 1/2 m₁ u₁ ²+ 1/2 m₂ u₂ ²= 1/2 m₁ v₂ ²+ 1/2 m₁ v₂ ² Examples

1. Collisions between billiard balls (approximated as elastic, as very little energy is lost).

2. Collisions between subatomic particles (often perfectly elastic).

Inelastic Collision

An inelastic collision is a collision in which momentum is conserved, but kinetic energy is NOT conserved. Some of the initial kinetic energy is converted into other forms of energy, such as heat, sound, or deformation (e.g., crumpling, breaking).

Figure 9.8: A diagram showing inelastic collision Characteristics

1. Total momentum is conserved.

2. Total kinetic energy is NOT conserved (it decreases).

3. Energy is transformed into other forms (heat, sound, deformation).

4. Objects may deform, or stick together, after the collision.

Mathematical Conditions

Conservation of momentum m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂ Examples of Inelastic Collisions (General)

1. Car crashes

2. A clay ball hitting a wall and sticking to it.

3. A dropped ball not bouncing back to its original height (some energy lost to heat and sound on impact).

Perfectly Inelastic Collision

This is a special and extreme case of an inelastic collision where the colliding objects stick together and move as a single combined mass after the collision.

This particular type of inelastic collision results in the maximum possible loss of kinetic energy (consistent with momentum conservation).

Figure 9.9: A diagram showing a perfectly inelastic collision

Example

A bullet embedding itself in a wooden block (ballistic pendulum), two cars colliding and sticking together, a supermarket trolley colliding with another from behind and then moving off together. In this specific case, the objects share a common final velocity.

m₁ u₁ + m₂ u₂ = ( m₁ + m₂)v

Activity 9.10 Verifying Newton’s Third Law

Objective: To verify Newton’s third law of motion by analysing momentum changes in a two-body elastic collision and demonstrating that the forces exerted by the bodies on each other are equal in magnitude and opposite in direction.

What you need

1. Calculator

2. Paper and pencil for calculations and notes

3. Data below Mass (kg) Initial Velocity u (m/s) Final Velocity v (m/s) m1 = 0.20 u1 = 6.0 v1 = 0.0 m² = 0.20 u² = 0.0 v² = 6.0 What to do

1. Record the masses m1 and m2, initial velocities u1 and u2, and final velocities v1 and v2.

2. Compute the initial momentum of each body: p₁ᵢ = m₁ u₁ and p₂ᵢ = m₂ u₂ Compute the final momentum of each body: p₁_(f) = m₁ v₁ and p₂ᵢ = m₂ v₂

3. Calculate Momentum Change for Each Body

∆p₁ = p₁_(f) − p₁ᵢ ∆p₂ = p₂_(f) − p₂ᵢ

4. Compare Momentum Changes

a. Check if the magnitude of the momentum changes satisfy:

∣Δp1∣=∣Δp2∣

b. Verify that the directions of momentum change are opposite (i.e., one gains momentum while the other loses an equal amount).

5. Discuss Newton’s Third Law

a. Explain how this equality and opposition of momentum changes illustrate Newton’s third law — for every action there is an equal and opposite reaction.

b. Discuss how the force during collision on body 1 by body 2 is equal in magnitude and opposite in direction to that on body 2 by body 1.

Activity 9.11 Exploring Elastic and Inelastic Collisions

Objective: To understand the difference between elastic and inelastic collisions by using the PhET Collision Lab simulation to compare how momentum and kinetic energy behave in each case.

What you need

1. Access to the PhET Collision Lab simulation https://phet.colorado.edu/en/simulation/collision-lab

2. Paper or notebook and pen or pencil What to do

1. Go to the PhET Collision Lab link above and select “Explore 1D” to work with one-dimensional collisions.

2. Set Up for Elastic Collisions

a. Check the boxes for “momentum” and “values” on the right menu to display numerical data.

b. Set the elasticity slider to 100% (for perfectly elastic collisions).

c. Set the masses for the two carts (balls) using either the sliders or by typing numbers, and assign initial velocities (e.g., one moving toward the other which is stationary).

3. Run and Record Data for Elastic Collisions

a. Press “play” to run the collision simulation.

b. Pause the simulation immediately after the collision.

c. Record the masses, velocities, momentum, and kinetic energy of each ball before and after the collision in a data table you create in your notebook.

d. Calculate total momentum and total kinetic energy before and after collision.

e. Observe and note if momentum and kinetic energy are conserved.

4. Set Up for Inelastic Collisions

a. Reset the simulation.

b. Move the elasticity slider to 0% (for perfectly inelastic collisions).

c. Repeat the process: set masses and initial velocities, run the simulation, pause after collision, and record data.

5. Run and Record Data for Inelastic Collisions

a. Fill out the data table with masses, velocities, momentum, and kinetic energy before and after collision.

b. Calculate total momentum and total kinetic energy before and after collision.

c. Notice how momentum is conserved but kinetic energy is not, due to energy loss (e.g., heat, deformation).

6. Explore Intermediate Elasticities and Mass Combinations (Optional)

a. Test collisions at intermediate elasticity values (e.g., 50%) or with unequal masses to observe varying behaviour.

b. Record and analyse corresponding data.

7. Summarise Findings Using Tables and Diagrams. Create a comparison

table listing:

a. Collision type (Elastic, Inelastic).

b. Whether momentum is conserved.

c. Whether kinetic energy is conserved.

d. Example observations or velocity changes.

Activity 9.12 Calculating Momentum, Impulse, and Velocities

Study the worked examples carefully before attempting the sample questions that follow.

Worked Example 1

Kudzo is going by a slippery snowy hill. He has a mass of 25kg, and he is sliding the hill at a velocity of 6m/s. Kudzo’s elder brother has a mass of 40kg.

His brother is moving slower with a velocity of 2m/s. Kudzo collides with his brother. Then both of them keep going down the hill as one unit. Calculate the resulting velocity of Kudzo and the impulse on his brother.

Solution

Mass of Kudzo m₁ = 25kg Mass of Kudzo’s brother m₂= 40kg Initial velocity of Kudzo u₁= 6m/s Initial velocity of Kudzo’s brother u₂= 2m/s Their common final velocity v =?

Since the collision is perfectly inelastic, then m₁ u₁ + m₂ u₂ = ( m₁ + m₂)v 25 × 6 + 40 × 2 = (25 + 40)v 150 + 80 = 65v 230 = 65v v = 3.538m / s Hence the resulting velocity of Kudzo is 3.538m/s.

The impulse on the brother I = m₂v – m₂u₂ I = 40 × 3.538 − 40 × 2 I = 141.52 − 80 I = 61.52 kgm / s

Worked Example 2

A block of mass m₁=2 kg moving with an initial velocity of u₁=+5 m/s collides elastically with a stationary block of mass m₂=3 kg. Find the final velocities of both blocks (note: there are two possible solutions to this question).

Solution

m₁= 2kg, u₁= 5, m₂= 3kg, u₂= 0 Conservation of Momentum m₁ u₁+m₂ u₂= m₁v₁+m₂ v₂ (2)(5) + (3)(0) = 2 v₁+ 3 v₂ 10 = 2 v₁+ 3 v₂ …………………………………………… (1) Conservation of Kinetic Energy 1__ 2m₁u₁ ²+ 1__ 2m₂ u₂ ²= 1__ 2m₁v₁ ²+ 1__ 2m₂ v₂ ²1/2(2) (5)²+ 1/2(3) (0)²= 1/2(2)v 1 2 + 1/2(3) v₂ ²25 = v₁ ²+ 1.5 v₂ ²………………………………………… (2) From eqn (1) v₁ = 10 − 3 v₂______ 2 , putting this into eqn (2) and expanding gives 100 = 100 − 60 v₂ + 15 v₂ ²v₂ ²− 4 v₂ = 0 v₂(v₂ − 4) = 0 v₂ = 0 v₂ = 4 Putting these values into v₁ = 10 − 3 v₂______ 2 gives v₁ = 10 − 3(0)_______ 2 v₁ = 5m / s Or v₁ = 10 − 3(4)_______ 2 v₁ = − 1m / s Hence the possible final velocities of block 1 and block 2 are v₁ = − 1m / s, 5m/s and v₂ = 0m / s, 4m / s respectively.

Practice Question

1. A 0.15 kg baseball is moving at 40 m/s when it is hit by a bat. The bat reverses the ball’s direction, and the ball leaves the bat with a speed of 50 m/s. The collision lasts for 0.002 seconds. Calculate the impulse delivered to the ball and the average force exerted on it by the bat.

2. A 5 kg block slides at 6 m/s on a frictionless surface and collides with a stationary 3 kg block. After the collision, the 5 kg block continues in the same direction at 2 m/s. What is the final velocity of the 3 kg block?

Activity 9.13 Real-world Examples of Momentum Conserved Collisions

Objective: To investigate real-world examples of collisions in order to explain how momentum is conserved and to distinguish between elastic and inelastic collisions.

What you need

1. Access to internet for online research

2. Access to videos, articles, or other multimedia resources about collisions in the real world

3. Paper or notebook for note-taking

4. Pen or pencil

5. Tools for creating presentations (e.g., PowerPoint, Google Slides, poster paper, or simple drawing tools)

6. Access to video creation or editing apps (optional, for short video clips) What to do

1. Form small groups of 3-5 people.

2. Each group selects one or more real-life examples of collisions. Some suggested examples include:

a. Car crashes (inelastic collisions where vehicles deform and stick/ slow down)

b. Billiard or pool ball collisions (elastic collisions where balls bounce off)

c. Sports collisions like tackles in football or collisions between cricket/ basketball balls

d. Collisions in machinery, such as two gears or parts hitting each other

e. Everyday events such as a mudball sticking to a wall or a ball bouncing on the ground

3. Use online resources, videos, or articles to gather information on how collisions occur in these examples, noting the type of collision (elastic or inelastic) and how momentum conservation applies.

4. For your chosen example, explain:

a. The principle of momentum conservation and how it applies to the collision

b. Whether the collision is elastic (kinetic energy conserved) or inelastic (some kinetic energy lost)

c. How velocity and mass influence the outcome of the collision

d. Any energy transformations involved (e.g., sound, heat, deformation)

5. Prepare support materials

a. Create diagrams illustrating the collision process, showing initial and final velocities, directions, and masses of the objects involved.

b. If possible, find or create short video clips demonstrating similar collisions.

c. Organise your findings into a clear and concise presentation format (slides, posters, or video).

6. Present to the class

a. Each group delivers a short presentation to the whole class or to one other group explaining their example, the momentum conservation principle involved, the collision type, and their supporting visuals or videos.

b. Explain how the laws of physics manifest in the real-world scenario you researched.

Activity 9.14 Videos of Different Types of Collision

What to do

1. Watch the video using the linked (or QR code) to observe example of different types of collision: Types of Collisions explained with animation

2. Write a brief summary of each example, explaining whether the collision is elastic or inelastic and how you know.

Activity 9.15 Egg Drop Experiment – Investigating Collision Time and Force Objective: To demonstrate how increasing the collision time during an impact reduces the force on an object (an egg), using different surface materials.

Materials

1. Raw eggs (at least 5–10)

2. Measuring tape or ruler

3. A step stool or safe elevated drop point (~1 metre height)

4. Surface materials (e.g.)

a. Concrete or tile (hard surface)

b. Foam pad or cushion

c. Bubble wrap

d. Sand or soil

e. Towel or thick fabric

5. Notebook or data sheet for observations Procedure

1. Set Up Surfaces

a. Place each surface material on the ground, spaced apart if possible.

b. Ensure each surface area is large enough to catch the falling egg.

2. Measure Drop Height: Use the measuring tape to ensure the drop height is consistent (e.g., 1 metre).

3. Drop the Egg

a. Carefully hold a raw egg at the set height.

b. Drop it directly onto the first surface.

c. Observe whether the egg breaks or remains intact.

4. Repeat

a. Use a new egg for each surface.

b. Drop from the same height onto each different material.

c. Record the result (e.g., broken/not broken, amount of cracking).

5. Analyse Results

a. Compare which surfaces allowed the egg to survive.

b. Discuss how softer or more flexible surfaces increase the collision time, reducing the force experienced by the egg (per the impulse- momentum principle).

Activity 9.16 Skateboard Experiment

Objective: To demonstrate and verify the conservation of momentum by observing the motion of two people pushing off each other while standing on skateboards.

Materials

1. 2 skateboards or rolling carts

2. 2 people of known (or measurable) mass

3. Measuring tape or marked floor (to track distance)

4. Stopwatch (optional)

5. Scale (to measure mass, if unknown)

6. Smooth, flat surface (e.g., gym floor) Procedure

1. Measure Masses. Record the mass of Person A and Person B.

2. Set Up: Both people stand still on their skateboards, facing each other, with hands touching lightly.

3. Push Off

a. Have the two people push off each other at the same time.

b. Allow them to roll freely in opposite directions.

4. Measure Velocities

a. Use a stopwatch and the marked floor to estimate each person’s velocity after the push.

b. Alternatively, use video analysis for more precise velocity measurements.

5. Calculate Momentum

a. Calculate momentum for each person.

b. Determine total momentum before (which should be 0, since both were at rest).

c. Add momenta after the push. The total momentum should still be approximately zero (accounting for direction, one will be negative).

6. Analysis: If momentum is conserved, the final momentum sum should be close to zero (within experimental error).