Cambridge IGCSE0972

Momentum

Physics 0972 Chapter Notes

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Momentum
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In this chapter

  1. 1Defining and Calculating Momentum
  2. 2The Principle of Conservation of Momentum
  3. 3Force as Rate of Change of Momentum
  4. 4Impulse and Its Role in Safety

1. Defining and Calculating Momentum

In physics, momentum is a measure of an object's 'quantity of motion'. It's a crucial concept for understanding collisions and interactions. Any object that has mass and is moving has momentum. It is defined as the product of the object's mass and its velocity. The formula is p = m × v, where p is momentum, m is mass, and v is velocity. Because velocity is a vector (it has both magnitude and direction), momentum is also a vector. We often use positive and negative signs to represent opposite directions along a straight line. The standard unit for momentum is kilogram-metres per second (kg m/s).

p = m × v

Key term

Momentum: The product of an object's mass and its velocity, a vector quantity measured in kg m/s.

Common pitfall

Forgetting that momentum is a vector. The direction is just as important as the magnitude, especially in collision problems.

Fun fact

An ocean liner moving at just a few miles per hour has enormous momentum due to its huge mass and would take several miles to stop. A bullet has large momentum due to its high velocity, despite its small mass.

Worked example 12 marks

A rugby player with a mass of 110 kg runs at a velocity of 8 m/s towards the try line. What is his momentum?

  1. 1

    State the formula for momentum: p = m × v

  2. 2

    Substitute the given 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 towards the try line.

Recap

  • Momentum is the product of mass and velocity (p = m × v).
  • The unit of momentum is kg m/s.
  • Momentum is a vector quantity, meaning it has both magnitude and direction.
  • An object must be moving to have momentum; a stationary object has zero momentum.

Quick check

  1. What are the standard units for momentum?1 mark
  2. A 5 kg object is stationary. What is its momentum?1 mark

2. 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 a system of interacting objects, the total momentum remains constant, provided no external forces (like friction) are acting. This means the total momentum before a collision or explosion is exactly equal to the total momentum after. We can write this as: Total momentum before = Total momentum after. When solving problems, we add up the individual momenta of all objects before the event and set it equal to the sum of their momenta after the event. Remember to use positive and negative signs to account for direction.

Total momentum before = Total momentum after

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Key term

Conservation of Momentum: The principle that in a closed system, the total momentum of all objects before an interaction (like a collision) is equal to the total momentum of all objects after the interaction.

Examiner insight

Examiners award marks for explicitly stating the principle of conservation of momentum and for setting up the 'total momentum before = total momentum after' equation correctly, paying close attention to signs for direction.

Worked example 14 marks

A 2 kg trolley moving at 3 m/s collides with a 4 kg stationary trolley. They stick together after the collision. What is their common velocity after the collision?

  1. 1

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

  2. 2

    Calculate total momentum before: p_before = (2 kg × 3 m/s) + (4 kg × 0 m/s) = 6 kg m/s.

  3. 3

    Define the system after the collision: The two trolleys move together with a combined mass of 2 kg + 4 kg = 6 kg. Let their common velocity be v.

  4. 4

    Calculate total momentum after: p_after = (6 kg) × v.

  5. 5

    Equate momentum before and after: 6 kg m/s = 6v.

  6. 6

    Solve for v: v = 6 / 6 = 1 m/s.

  7. 7

    The common velocity of the trolleys is 1 m/s in the original direction of the 2 kg trolley.

Worked example 23 marks

An astronaut of mass 80 kg is stationary in space. She throws a 2 kg spanner away from her at a velocity of 5 m/s. What is her recoil velocity?

  1. 1

    The system is the astronaut and the spanner. Initially, everything is stationary, so the total initial momentum is 0.

  2. 2

    State the principle of conservation of momentum: Total momentum before = Total momentum after. So, 0 = Total momentum after.

  3. 3

    Let the astronaut's recoil velocity be v. Let the spanner's direction be positive, so its velocity is +5 m/s. The astronaut will move in the opposite direction.

  4. 4

    Total momentum after = (momentum of astronaut) + (momentum of spanner) = (80 kg ×v) + (2 kg × 5 m/s).

  5. 5

    Set total momentum after to zero: 80v + 10 = 0.

  6. 6

    Solve for v: 80v = -10, so v = -10 / 80 = -0.125 m/s.

  7. 7

    The negative sign indicates her velocity is in the opposite direction to the spanner. Her recoil speed is 0.125 m/s.

Recap

  • Total momentum is conserved in any collision or explosion if no external forces act.
  • Total momentum before an event equals total momentum after the event.
  • Always define a direction as positive and use negative signs for the opposite direction.
  • In 'explosion' problems where objects start from rest, the initial momentum is zero.
  • In 'sticking' collisions, the final mass is the sum of the individual masses.

Quick check

  1. Under what condition is the total momentum of a system conserved?1 mark

3. Force as Rate of Change of Momentum

Newton's Second Law of Motion is often stated as F = ma, but its more fundamental form relates force to momentum. It states that the resultant force acting on an object is equal to the rate of change of its momentum. A force is required to change an object's momentum, and a larger force will cause a faster change. The equation is: Resultant Force = (change in momentum) / (time taken). The change in momentum is the final momentum minus the initial momentum (Δp = mv - mu).

F = Δp / Δt

F = (mv - mu) / t

Key term

Rate of change of momentum: The change in an object's momentum divided by the time taken for the change to occur, which is equal to the resultant force on the object.

Examiner insight

Marks are often awarded for correctly calculating the change in momentum (final momentum - initial momentum), especially when the object reverses direction. In that case, if initial velocity is u, final velocity is -v, so the change in momentum is m(-v) - mu = -m(v+u).

Worked example 13 marks

A 0.5 kg football is travelling at 10 m/s. A player kicks it, and 0.1 seconds later it is travelling at 20 m/s in the same direction. What is the average resultant force exerted by the player on the ball?

  1. 1

    Calculate the initial momentum: p_initial = m × u = 0.5 kg × 10 m/s = 5 kg m/s.

  2. 2

    Calculate the final momentum: p_final = m × v = 0.5 kg × 20 m/s = 10 kg m/s.

  3. 3

    Calculate the change in momentum: Δp = p_final - p_initial = 10 - 5 = 5 kg m/s.

  4. 4

    Use the formula for force: F = Δp / Δt.

  5. 5

    Substitute the values: F = 5 kg m/s / 0.1 s.

  6. 6

    Calculate the force: F = 50 N.

Worked example 24 marks

A 150 g cricket ball hits a bat at 30 m/s and is hit straight back towards the bowler at 40 m/s. The ball is in contact with the bat for 0.05 s. Calculate the average force the bat exerts on the ball.

  1. 1

    Convert mass to kg: 150 g = 0.15 kg. Define initial direction as positive, so u = +30 m/s. The final velocity is in the opposite direction, so v = -40 m/s.

  2. 2

    Calculate initial momentum: p_initial = 0.15 kg × 30 m/s = 4.5 kg m/s.

  3. 3

    Calculate final momentum: p_final = 0.15 kg × (-40 m/s) = -6.0 kg m/s.

  4. 4

    Calculate the change in momentum: Δp = p_final - p_initial = (-6.0) - (4.5) = -10.5 kg m/s.

  5. 5

    Use the force formula: F = Δp / Δt = -10.5 kg m/s / 0.05 s.

  6. 6

    Calculate the force: F = -210 N. The negative sign shows the force is in the opposite direction to the ball's initial motion. The magnitude is 210 N.

Recap

  • A resultant force causes a change in momentum.
  • Force is equal to the rate of change of momentum.
  • The formula is F = (mv - mu) / t.
  • Be extremely careful with signs when an object changes direction.

Quick check

  1. State Newton's second law in terms of momentum.1 mark

4. Impulse and Its Role in Safety

Impulse is defined as the product of a force and the time for which it acts. It is also equal to the change in momentum it produces. The formula is Impulse = F × t = Δp. This relationship is vital for designing safety features. In any situation where momentum changes, like a crash, the change in momentum (Δp) is fixed. To reduce the force (F) experienced, we must increase the time(t) over which the force acts. This is the principle behind airbags, crumple zones in cars, cushioned flooring in gyms, and bending your knees when you land from a height. They all work by extending the collision time to reduce the peak force, thus minimising injury.

Impulse = F × t

F × t = Δp

Key term

Impulse: The product of a force and the time for which it acts, which is equal to the change in momentum it produces (measured in Ns or kg m/s).

Examiner insight

When explaining safety features, to get full marks you must state that the feature increases the time of impact, which for a fixed change in momentum, decreases the resultant force according to the impulse-momentum equation.

Fun fact

Stunt performers jumping from buildings land on giant airbags. The bag deflates slowly on impact, increasing the time to stop their momentum from a large value to zero, which keeps the force on their body below a fatal level.

Worked example 15 marks

A car of mass 1200 kg hits a wall and stops. Its initial velocity was 15 m/s.(a) What is the change in momentum of the car?(b) If the car has a crumple zone that allows it to stop in 0.3 s, what is the average force on the car?(c) If the car were rigid and stopped in 0.05 s, what would be the force?

  1. 1

    (a) Initial momentum p_i = 1200 kg × 15 m/s = 18000 kg m/s. Final momentum p_f = 0. Change in momentum Δp = p_f - p_i = 0 - 18000 = -18000 kg m/s.

  2. 2

    (b) Use F = Δp / t. F = -18000 kg m/s / 0.3 s = -60,000 N. The average force is 60,000 N.

  3. 3

    (c) Use F = Δp / t. F = -18000 kg m/s / 0.05 s = -360,000 N. The average force is 360,000 N, which is six times larger.

Recap

  • Impulse is force multiplied by time (F × t).
  • Impulse is equal to the change in momentum (Δp).
  • For a fixed change in momentum, increasing the impact time decreases the impact force.
  • Safety features like airbags and crumple zones work by increasing the impact time.

Quick check

  1. Explain in one sentence why bending your knees when you land from a jump reduces the risk of injury.2 marks

End-of-chapter exercise

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

  1. A bowling ball of mass 7 kg travels at a velocity of 4 m/s. Calculate its momentum.2 marks
  2. A force of 20 N acts on an object for 3 seconds. What is the impulse delivered to the object?2 marks
  3. A 1000 kg car travelling at 20 m/s collides with a stationary 1500 kg van. The two vehicles lock together. Calculate their velocity immediately after the collision.4 marks
  4. A 0.2 kg ball is dropped from rest. After falling for 2 s, it has a velocity of 20 m/s (ignore air resistance). Calculate the change in momentum of the ball.2 marks
  5. A machine gun fires 300 bullets per minute. Each bullet has a mass of 10 g (0.01 kg) and leaves the gun with a velocity of 500 m/s. Calculate the average recoil force on the gun.4 marks
  6. Explain, using the concepts of momentum and impulse, how a car's airbag can prevent serious head injuries in a collision.3 marks
  7. A 5 kg trolley moving to the right at 4 m/s collides with a 2 kg trolley moving to the left at 3 m/s. After the collision, the 2 kg trolley moves to the right at 2 m/s. What is the velocity of the 5 kg trolley after the collision?5 marks
  8. A tennis player hits a 60 g (0.06 kg) ball which approaches the racket at 50 m/s. The ball is in contact with the racket for 5 milliseconds (0.005 s) and leaves in the opposite direction at 40 m/s. Calculate the magnitude of the average force exerted by the racket on the ball.4 marks
  9. A 70 kg person jumps from a wall. Just before they hit the ground, their speed is 6 m/s. They bend their knees, bringing them to a stop in 0.5 s. Calculate the average resultant force on the person while they are stopping.3 marks
  10. Two ice skaters, A (mass 60 kg) and B (mass 80 kg), are stationary on the ice. They push off from each other. Skater A moves away with a velocity of 2 m/s. a) What is the total momentum of the system before they push off? b) What is the velocity of skater B after they push off? c) In this interaction, is kinetic energy conserved? Explain your answer.6 marks

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