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CCEA GCE Mathematics (2210) · Impulse and momentum

Mini-Lesson

Impulse and momentum

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Mission 1A 1 kg trolley at 4 m s⁻¹ hits a still 3 kg one. They stick. How fast?

The specification points and sources are listed under the lesson: see Sources.

Mission 1

A 1 kg trolley at 4 m s⁻¹ hits a still 3 kg one. They stick. How fast?

Make a guess: A 1 kg trolley at 4 m s⁻¹ hits a still 3 kg one. They stick. How fast?

Mechanics · impulse and momentum (A2 only)

Momentum and impulse

Momentum $= m v$ =mv. It is a vector: in one dimension, choose a positive direction and give velocities signs. Units: N s (or kg m s⁻¹).
Impulse of a constant force $F$ F acting for time $t$ t is $Ft$ Ft, in N s.
Impulse = change in momentum:

I = m v - m u

The impulse has the same direction as the change in velocity.

Worked example Mechanics · impulse and momentum (A2 only)

A ball rebounding from a wall

Question

A ball of mass 0.2 kg hits a vertical wall at 15 m s⁻¹ and rebounds along the same line at 10 m s⁻¹. Find the impulse on the ball, and the average force if contact lasts 0.04 s.

1
Take towards the wall as positive: $u = 15$ u=15, $v = - 10$ v=−10
2
Use $I = m v - m u$ I=mv−mu: $I = 0.2 (- 10) - 0.2 (15) = - 5$ I=0.2(−10)−0.2(15)=−5
3
Interpret the sign: 5 N s away from the wall
4
Average force: $F = \frac{I}{t} = \frac{5}{0.04} = 125$ F=I/t=5/0.04=125

Answer

impulse 5 N s away from the wall; average force 125 N

⚠️ Watch out

0.2 (15 - 10) = 1

N s ignores the reversal; the velocity changes by 25 m s⁻¹, not 5.

Calculate

?A ball of mass 0.4 kg moving at 6 m s⁻¹ is hit by a bat and moves back along the same line at 14 m s⁻¹. Find the magnitude of the impulse on the ball.

N s

Mechanics · impulse and momentum (A2 only)

Conservation of linear momentum

When two bodies collide, they exert equal and opposite impulses on each other (Newton's third law).
So, with no external impulse, total momentum before = total momentum after:

m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}

Coalesce: the bodies stick and share one velocity, $(m_{1} + m_{2}) v$ (m₁+m₂)v.
Explosion from rest: total momentum is zero before, so the parts move in opposite directions.

Worked example Mechanics · impulse and momentum (A2 only)

A head-on collision

Question

Particle

A

(3 kg) moving at 5 m s⁻¹ collides directly with particle

B

(2 kg) moving at 4 m s⁻¹ towards

A

. After the collision

A

moves at 1 m s⁻¹ in its original direction. Find the velocity of

B

after the collision.

1
Take $A$ A's original direction as positive, so $B$ B's velocity is $- 4$ −4
2
Momentum before: $3 (5) + 2 (- 4) = 7$ 3(5)+2(−4)=7
3
Momentum after: $3 (1) + 2 v_{B} = 7$ 3(1)+2v_B=7, so $v_{B} = 2$ v_B=2
4
Check impulses: on $A$ A, $3 (1 - 5) = - 12$ 3(1−5)=−12; on $B$ B, $2 (2 - (- 4)) = 12$ 2(2−(−4))=12

Answer

B

moves at 2 m s⁻¹ in

A

's original direction (it reverses)

⚠️ Watch out

writing

B

's initial velocity as

+ 4

gives

v_{B} = 10

, which is wrong.

Quick check

?A shell of mass 5 kg is at rest. It explodes into two parts of mass 2 kg and 3 kg. The 2 kg part moves off at 12 m s⁻¹. Find the speed of the 3 kg part.

Calculate

?Particle

A

(4 kg) moving at 6 m s⁻¹ collides directly with particle

B

(

m

kg) moving at 3 m s⁻¹ towards

A

. They coalesce and move at 1.5 m s⁻¹ in

A

's original direction. Find

m

Calculate

?Hence find the magnitude of the impulse exerted on

A

in this collision.

N s

Mission check

?A 1 kg trolley at 4 m s⁻¹ hits a stationary 3 kg trolley and they stick together. How fast do they move?

Recap

Big ideas

Momentum $= m v$ =mv (a vector); impulse $= Ft = m v - m u$ =Ft=mv−mu, in N s.
With no external impulse, total momentum is conserved in collisions, coalescence and explosions.
Choose a positive direction first and give every velocity its sign.

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Mission 1A 1 kg trolley at 4 m s⁻¹ hits a still 3 kg one. They stick. How fast?

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Specification points and sources for this lesson are listed at the bottom of the page.

Sources (1)

The specification points and sources behind this lesson.

1 CCEA GCE Mathematics (2018) specification, Unit A2 2 Applied Mathematics, Section A Mechanics, Impulse and momentum