Momentum & Impulse (Cambridge O Level Physics)

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Momentum

  • An object with mass that is in motion has momentum which is defined by the equation:

momentum = mass × velocity

p = mv

  • Where:
    • p = momentum in kilogram metre per second (kg m/s)
    • m = mass in kilograms (kg)
    • v = velocity in metres per second (m/s)

  • This means that an object at rest (i.e v = 0) has no momentum
  • Momentum keeps an object moving in the same direction, making it difficult to change the direction of an object with a large momentum
  • Since velocity is a vector this means that the momentum of an object also depends on its direction of travel
  • This means that momentum can be either positive or negative
    • If an object travelling to the right has positive momentum, an object travelling in the opposite direction (to the left) will have negative momentum

Momentum of a Tennis Ball

Negative momentum, downloadable AS & A Level Physics revision notes

The tennis ball's momentum is negative when it moves in the opposite direction to which it initially was travelling in

  • Therefore, the momentum of an object will change if:
    • The object accelerates (speeds up) or decelerates (slows down)
    • Changes direction
    • Its mass changes

Worked example

Which object has the most momentum?

WE - Momentum comparison question image, downloadable AS & A Level Physics revision notes

Answer:

  • Momentum of tennis ball

p space equals space m v

p space equals space 0.06 space cross times space 75

p space equals space 4.5 space kg space straight m divided by straight s

  • Momentum of brick

p space equals space m v

p space equals space 3 space cross times space 1.5

p space equals space 4.5 space kg space straight m divided by straight s

  • Both the tennis ball and the brick have the same momentum
  • Even though the brick is much heavier than the ball, the ball is travelling much faster than the brick
  • This means that on impact, they would both exert a similar force (depending on the time it takes for each to come to rest)

Examiner Tip

You can remember momentum as mass in motion. The units of momentum are kg m/s which is the product of the units of mass (kg) and velocity (m/s).

Which direction is taken as positive is completely up to you in the exam. In general, the right and upwards are taken as positive, and down or to the left as negative.

Conservation of Momentum

  • The principle of conservation of momentum states that:

In a closed system, the total momentum before an event is equal to the total momentum after the event

  • A closed system means the energy within the system is constant and there is an absence of external forces (e.g. friction)
  • In other words:

The total momentum before a collision = The total momentum after a collision

  • A system is a certain number of objects under consideration
    • This can be just one object or multiple objects

  • Since momentum is a vector quantity, a system of objects moving in opposite directions (e.g. towards each other) at the same speed will have an overall momentum of 0 since they will cancel out
    • Momentum is always conserved over time

  • The diagram below shows two masses with velocity u and M at rest (ie. zero velocity)

Momentum of Masses Before and After a Collision

conservation-of-momentum, IGCSE & GCSE Physics revision notes

The momentum of a system before and after a collision

  • Before the collision:
    • The momentum is only of mass m which is moving
    • If the right is taken as the positive direction, the total momentum of the system is m × u

  • After the collision:
    • Mass M also now has momentum
    • The velocity of m is now -(since it is now travelling to the left) and the velocity of M is V
    • The total momentum is now the momentum of M + momentum of m
    • This is (M × V) + (m × -v) or (M × V) – (m × v)

Worked example

The diagram shows a car and a van, just before and just after the car collided with the van, which is initially at rest.

WE Conservation of Momentum Question image, downloadable IGCSE & GCSE Physics revision notes

Use the idea of conservation of momentum to calculate the velocity of the van when it is pushed forward by the collision.

Answer:

Step 1: State the principle of conservation of momentum

total space momentum space before space equals space total space momentum space after

Step 2: Calculate the total momentum before the collision

p space equals space m v

  • Momentum of the car

p space equals space 990 space cross times space 10

p space equals space 9900 space kg space straight m divided by straight s

  • Momentum of the van
    • The van is at rest
    • Therefore, v = 0 m/s
    • Therefore, p = 0 kg m/s
  • Total momentum before collision

p subscript b e f o r e end subscript space equals space 9900 space plus space 0 space equals space 9900 space kg space straight m divided by straight s

Step 3: Calculate the total momentum after the collision

  • Momentum of the car

p space equals space 990 space cross times space 2

p space equals space 1980 space kg space straight m divided by straight s

  • Momentum of the van

p space equals space 4200 space cross times space v

  • Total momentum after collision

p subscript a f t e r end subscript space equals space 1980 space plus space 4200 v

Step 4: Rearrange the conservation of momentum equation to solve for v

p subscript b e f o r e end subscript space equals space p subscript a f t e r end subscript

9900 space equals space 1980 space plus space 4200 v

9900 space minus space 1980 space equals space 4200 v

v space equals space fraction numerator 9900 space minus space 1980 over denominator 4200 end fraction

v space equals space 1.9 space straight m divided by straight s

Examiner Tip

If it is not given in the question already, drawing a diagram of before and after helps keep track of all the masses and velocities (and directions) in the conversation of momentum questions.

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Leander

Author: Leander

Expertise: Physics

Leander graduated with First-class honours in Science and Education from Sheffield Hallam University. She won the prestigious Lord Robert Winston Solomon Lipson Prize in recognition of her dedication to science and teaching excellence. After teaching and tutoring both science and maths students, Leander now brings this passion for helping young people reach their potential to her work at SME.