Maths › Further Mechanics 1 › Momentum and impulse
Momentum and impulse
Force applied for a time changes momentum by exactly that much. When two bodies push on each other the pushes are equal and opposite, so the total momentum cannot change at all.
Builds on Forces and Newton's laws and Kinematics with constant acceleration.
IN THIS TOPIC
- Use impulse equals change in momentum, including for a rebound.
- Apply conservation of momentum to a direct collision between two spheres.
- Handle coalescence and explosion as special cases of the same principle.
WHAT YOU PROBABLY THINK
A ball that bounces back off a wall at the same speed it arrived receives no impulse, since its speed is unchanged.
Force for a time
Momentum is mass times velocity, a vector measured in newton seconds. Newton's second law rearranges into the impulse-momentum principle:
Because momentum is a vector, a rebound is a change of sign, and the change is the sum of the two speeds rather than their difference. The opening claim confuses speed with velocity: a ball arriving at 20 and leaving at 20 the other way has had its momentum reversed, which takes a large impulse.
For a force that varies, the impulse is the area under the force-time graph, and dividing it by the contact time gives the average force. That is why a longer contact means a gentler force for the same change of momentum, which is the whole idea behind crumple zones and follow-through in a stroke.
WORKED EXAMPLE
A struck ball
A ball of mass 0.15 kg arrives at 20 m/s and is struck straight back at 25 m/s. The contact lasts 0.02 s. Find the impulse and the average force.
Taking the outgoing direction as positive: u = −20, v = 25.
I = 0.15(25) − 0.15(−20) = 3.75 + 3 = 6.75 N s.
Average force = 6.75/0.02 = 337.5 N, over two hundred times the ball's weight.
Why the total cannot change
During a collision each body exerts a force on the other, and by Newton's third law those forces are equal and opposite for the same length of time. The impulses are therefore equal and opposite, so the momentum gained by one is exactly the momentum lost by the other. The total is conserved, whatever happens to the energy.
Coalescence, where the two move off together, and explosion, where one body separates into two, are the same equation with the masses grouped differently. Signs matter throughout: choose a positive direction, write every velocity with its sign, and let the algebra report which way anything is moving afterwards.
YOUR TURN
A direct collision
A sphere of mass 4 kg moving at 6 m/s meets a sphere of mass 3 kg moving at 2 m/s towards it. After the collision the 4 kg sphere continues in its original direction at 1 m/s. Find the velocity of the other sphere.
Show the working
Take the 4 kg sphere's direction as positive, so u₁ = 6 and u₂ = −2.
Total momentum before = 4(6) + 3(−2) = 24 − 6 = 18 N s.
After: 4(1) + 3v = 18, so 3v = 14 and v = 14/3 ≈ 4.67 m/s.
The sign is positive, so the 3 kg sphere has reversed and now moves in the original direction of the heavier one.
THE EXAM BIT
- Choose a positive direction, mark it on your diagram, and write every velocity with its sign.
- Impulse is a vector: a rebound adds the two speeds rather than subtracting them.
- Quote the units: N s for impulse and momentum, which is the same as kg m/s.
- For coalescence, use the combined mass on the right-hand side and one velocity.
CHECK YOURSELF
A particle of mass 3 kg moving at 5 m/s collides with a stationary particle of mass 2 kg and they move off together. Find their common speed.
Show a hint
Total momentum before equals combined mass times common velocity.
Show the answer
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Impulse is force times time and equals the change in momentum, so a rebound needs an impulse equal to the sum of the two speeds.
In any collision the internal forces are equal and opposite, so the total momentum before equals the total momentum after.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Use impulse equals change in momentum, including for a rebound.
- Apply conservation of momentum to a direct collision between two spheres.
- Handle coalescence and explosion as special cases of the same principle.
Open the full revision checklist to see every objective in the course in one place.
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