Cambridge IGCSE0625

Momentum

Physics 0625 Chapter Notes

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Momentum
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In this chapter

  1. 1Defining and Calculating Momentum
  2. 2Impulse and Changing Momentum
  3. 3The Principle of Conservation of Momentum
  4. 4Force as Rate of Change of Momentum

1. Defining and Calculating Momentum

In physics, momentum is not just a vague idea of 'unstoppability'; it's a precise quantity that all moving objects have. It's often described as the 'quantity of motion'. An object's momentum depends on two things: its mass (how much 'stuff' it's made of) and its velocity (how fast it's moving and in what direction). A heavy truck moving fast has a huge momentum, while a stationary object has zero momentum. The formula is simply momentum = mass × velocity. Because velocity has a direction, momentum is also a vector quantity. We must consider its direction, often by assigning positive values for one direction (e.g., right) and negative for the opposite (e.g., left).

p = m × v

Key term

Momentum (p): The product of an object's mass and its velocity; a vector quantity measured in kilogram metres per second (kg m/s).

Examiner insight

Examiners award marks for correctly stating the unit of momentum as kg m/s. Do not write kg/m/s or other incorrect variations.

Common pitfall

Forgetting that momentum is a vector quantity. In calculations involving changes of direction, you must use positive and negative signs to represent the different directions.

Worked example 12 marks

A rugby player with a mass of 110 kg runs at a velocity of 8 m/s to the right. Calculate his momentum.

  1. 1

    State the formula for momentum: p = m × v

  2. 2

    Substitute the known values: p = 110 kg × 8 m/s

  3. 3

    Calculate the result: p = 880 kg m/s

  4. 4

    State the final answer with units and direction: The player's momentum is 880 kg m/s to the right.

Worked example 23 marks

A ball of mass 0.5 kg moves to the right at 4 m/s. It hits a wall and bounces back, moving to the left at 3 m/s. What is its change in momentum?

  1. 1

    Define 'right' as the positive direction. Initial velocity(u) = +4 m/s. Final velocity(v) = -3 m/s.

  2. 2

    Calculate initial momentum (p_initial): p_initial = m × u = 0.5 kg × 4 m/s = +2.0 kg m/s.

  3. 3

    Calculate final momentum (p_final): p_final = m × v = 0.5 kg × (-3 m/s) = -1.5 kg m/s.

  4. 4

    Calculate the change in momentum (Δp): Δp = p_final - p_initial = (-1.5 kg m/s) - (2.0 kg m/s) = -3.5 kg m/s.

  5. 5

    The change in momentum is 3.5 kg m/s to the left.

Recap

  • Momentum is a measure of an object's quantity of motion.
  • The formula for momentum is p = m × v.
  • The standard unit for momentum is kg m/s.
  • Momentum is a vector quantity, meaning it has both magnitude and direction.
  • An object that is not moving has zero momentum.
  • The direction of momentum is the same as the direction of the velocity.

Quick check

  1. What is the momentum of a 60 kg person running at 3 m/s?1 mark
  2. Why is momentum considered a vector quantity?1 mark

2. Impulse and Changing Momentum

To change an object's momentum, you need to apply a resultant force. The longer that force acts, and the larger the force, the greater the change in momentum will be. This combination of force and time is called impulse. Impulse is defined as the product of the force and the time interval over which it acts. Crucially, the impulse applied to an object is exactly equal to the change in momentum it causes. This principle is vital in designing safety features. For example, car airbags and crumple zones increase the time of impact during a crash. For a given change in momentum (from high speed to zero), increasing the time (Δt) reduces the peak force (F), making the crash more survivable.

Impulse = F × Δt

Impulse = Δp (change in momentum)

F × Δt = mv - mu

Key term

Impulse: The change in momentum of an object, which is equal to the product of the resultant force acting on it and the time for which the force acts.

Common pitfall

Confusing impulse with force. Impulse is the overall effect of a force acting over a period of time (the 'punch'), not just the force itself.

Fun fact

When a gymnast lands on a mat, they bend their knees to increase the time over which their momentum changes to zero, reducing the force on their joints and preventing injury.

Worked example 14 marks

A footballer kicks a 0.45 kg ball, initially at rest. His boot is in contact with the ball for 0.05 s, and the ball leaves his foot with a velocity of 20 m/s. Calculate the average force exerted by the boot on the ball.

  1. 1

    State the formula linking force, time, and momentum change: F × Δt = mv - mu.

  2. 2

    Identify the values: m = 0.45 kg, u = 0 m/s, v = 20 m/s, Δt = 0.05 s.

  3. 3

    Calculate the change in momentum (Δp): Δp = (0.45 kg × 20 m/s) - (0.45 kg × 0 m/s) = 9.0 kg m/s.

  4. 4

    Rearrange the formula to find the force: F = Δp / Δt.

  5. 5

    Substitute the values: F = 9.0 kg m/s / 0.05 s = 180 N.

  6. 6

    The average force exerted is 180 N.

Recap

  • A resultant force is required to change an object's momentum.
  • Impulse is the product of force and the time for which it acts (Impulse = F × Δt).
  • Impulse is equal to the change in momentum (Δp).
  • The formula F × Δt = mv - mu links force, time and momentum change.
  • Safety features like airbags work by increasing the impact time to reduce the impact force.

Quick check

  1. A force of 50 N acts on an object for 0.2 s. What is the impulse given to the object?1 mark
  2. How do crumple zones in cars make them safer in a collision?2 marks

3. The Principle of Conservation of Momentum

One of the most fundamental laws in physics is the Principle of Conservation of Momentum. It states that for any interaction between objects in a closed system (meaning no external forces like friction or air resistance are acting), the total momentum before the interaction is equal to the total momentum after the interaction. Momentum is not lost or gained, only transferred between objects. This principle applies to all interactions, including collisions (where objects hit each other) and explosions (where objects push away from each other). By calculating the total momentum before and after, we can solve for unknown quantities like the velocity of an object after a crash.

Total momentum before = Total momentum after

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Key term

Conservation of Momentum: A principle stating that in a closed system, the total momentum before an event (like a collision) is equal to the total momentum after the event.

Examiner insight

Examiners look for a clear statement of the conservation principle and separate, well-organised calculations for the 'before' and 'after' states. Setting out your work clearly will gain you marks.

Common pitfall

In conservation of momentum problems, forgetting to add the masses of objects that stick together after a collision.

Worked example 14 marks

A 1000 kg car travelling at 20 m/s collides with a stationary 1500 kg van. After the collision, the two vehicles lock together. Calculate their common velocity immediately after the collision.

  1. 1

    State the principle: Total momentum before = Total momentum after.

  2. 2

    Calculate total momentum before: p_before = (m_car × u_car) + (m_van × u_van) = (1000 kg × 20 m/s) + (1500 kg × 0 m/s) = 20000 kg m/s.

  3. 3

    Define the 'after' state: The total mass is m_total = 1000 kg + 1500 kg = 2500 kg. Let their common velocity be 'v'.

  4. 4

    Calculate total momentum after: p_after = m_total × v = 2500 kg × v.

  5. 5

    Equate the two: 20000 kg m/s = 2500 kg × v.

  6. 6

    Solve for v: v = 20000 / 2500 = 8 m/s.

  7. 7

    The common velocity of the vehicles is 8 m/s in the original direction of the car.

Worked example 24 marks

A 70 kg skateboarder at rest on a 2 kg skateboard throws a 3 kg ball forwards with a velocity of 5 m/s. Calculate the recoil velocity of the skateboarder and skateboard.

  1. 1

    This is an 'explosion' type problem. The system is initially at rest.

  2. 2

    State the principle: Total momentum before = Total momentum after.

  3. 3

    Calculate total momentum before: p_before = 0 kg m/s (since everything is at rest).

  4. 4

    Therefore, the total momentum after must also be 0.

  5. 5

    Calculate momentum of the ball: p_ball = m_ball × v_ball = 3 kg × 5 m/s = +15 kg m/s (taking 'forwards' as positive).

  6. 6

    The skateboarder and board (total mass = 70+2=72 kg) must have an equal and opposite momentum. Let their velocity be 'v_skater'.

  7. 7

    p_after = p_ball + p_skater = 0. So, (+15 kg m/s) + (72 kg × v_skater) = 0.

  8. 8

    Solve for v_skater: 72 × v_skater = -15. So, v_skater = -15 / 72 ≈ -0.21 m/s.

  9. 9

    The recoil velocity is 0.21 m/s in the opposite direction to the ball.

Recap

  • In a closed system, total momentum is always conserved.
  • Total momentum before an interaction equals total momentum after.
  • This principle applies to both collisions and explosions.
  • Remember to treat momentum as a vector, using signs for direction.
  • For explosions from rest, the total final momentum is zero.

Quick check

  1. If two objects collide and stick together, is the total momentum of the system conserved?1 mark
  2. A cannon fires a cannonball forwards. In which direction does the cannon move, and why?2 marks

4. Force as Rate of Change of Momentum

You know Newton's Second Law as F = ma. However, its more fundamental and complete definition is that the resultant force on an object is equal to the rate of change of its momentum. This means F = (change in momentum) / (time taken). We can show how this leads to the familiar F = ma. Change in momentum is final momentum (mv) minus initial momentum (mu). So, F = (mv - mu) / t. Factoring out the mass gives F = m(v -u) / t. Since acceleration 'a' is defined as the rate of change of velocity, a = (v -u) / t, we arrive back at F = ma. This more complete definition is powerful because it works even when mass changes, such as in a rocket which burns fuel and gets lighter.

F = Δp / Δt

F = (mv - mu) / t

Key term

Newton's Second Law (in terms of momentum): The resultant force acting on an object is directly proportional to the rate of change of its momentum, and acts in the same direction.

Examiner insight

Students who can state and use Newton's Second Law in terms of momentum demonstrate a superior understanding of dynamics. This is often a key differentiator for top grades.

Fun fact

Rocket propulsion is a perfect example of F = Δp/Δt. The rocket expels hot gas (a mass of gas, Δm, with velocity v) in a time Δt, creating a change in momentum for the gas. This produces a large thrust force F on the rocket in the opposite direction.

Worked example 13 marks

Water from a hose strikes a wall at a rate of 5 kg per second. The water is moving at 10 m/s when it hits the wall and stops dead. Calculate the average force exerted on the wall by the water.

  1. 1

    State Newton's Second Law in terms of momentum: F = Δp / Δt.

  2. 2

    Consider a time interval of 1 second (Δt = 1 s). In this time, the mass of water hitting the wall is m = 5 kg.

  3. 3

    The initial velocity is u = 10 m/s and the final velocity is v = 0 m/s.

  4. 4

    Calculate the change in momentum (Δp) for this mass of water: Δp = mv - mu = (5 kg × 0 m/s) - (5 kg × 10 m/s) = -50 kg m/s.

  5. 5

    The change in momentum of the water is -50 kg m/s. By Newton's Third Law, the impulse on the wall is equal and opposite, so the change in momentum imparted to the wall is +50 kg m/s.

  6. 6

    Calculate the force: F = Δp / Δt = 50 kg m/s / 1 s = 50 N.

  7. 7

    The average force on the wall is 50 N.

Recap

  • Newton's Second Law states that force is the rate of change of momentum.
  • The formula is F = Δp / Δt or F = (mv - mu) / t.
  • This is a more complete definition than F = ma.
  • It explains how a continuous force is exerted by something like a jet of water.

Quick check

  1. State Newton's Second Law of Motion in terms of momentum.1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A bowling ball has a mass of 7.0 kg and travels at a velocity of 4.0 m/s. Calculate its momentum.2 marks
  2. State the principle of conservation of momentum and specify the condition under which it applies.2 marks
  3. A 1200 kg car travelling at 15 m/s collides head-on with a 800 kg car travelling at 10 m/s in the opposite direction. They lock together. Calculate their velocity immediately after the collision, stating the direction.4 marks
  4. Explain, using the concepts of momentum and impulse, how a car's airbag can reduce injury to a driver in a crash.3 marks
  5. A 2000 kg cannon is at rest on a horizontal surface. It fires a 10 kg cannonball horizontally with a speed of 80 m/s. Calculate the recoil speed of the cannon.3 marks
  6. A tennis ball of mass 60 g travelling at 40 m/s is hit by a racket. It returns in the opposite direction with a speed of 30 m/s. The ball is in contact with the racket for 50 ms. Calculate the average force exerted by the racket on the ball.5 marks
  7. A 60 kg ice skater and a 40 kg ice skater are at rest on the ice. They push each other apart. The 60 kg skater moves away with a speed of 2 m/s. What is the speed of the 40 kg skater?3 marks
  8. A 1.0 kg ball is dropped from a height of 2.0 m. It bounces back up to a height of 1.2 m. Assuming g = 10 m/s², calculate the change in momentum of the ball during its collision with the floor.5 marks
  9. Define impulse and state the equation that links it to the change in momentum.2 marks
  10. A machine gun fires 200 bullets per minute. Each bullet has a mass of 20 g and a speed of 500 m/s. Calculate the average force required to hold the gun steady.4 marks

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