Maths › Further Mechanics 1 › Direct impact and Newton's law of restitution
Direct impact and Newton's law of restitution
Conservation of momentum gives one equation and two unknowns. The coefficient of restitution supplies the second, and between them they settle any direct collision.
Builds on Momentum and impulse and Work, energy and power.
IN THIS TOPIC
- State Newton's law of restitution and the range of values e can take.
- Solve a direct collision using conservation of momentum and restitution together.
- Calculate the kinetic energy lost in an impact.
WHAT YOU PROBABLY THINK
Since momentum is conserved in a collision, energy is conserved too.
Two equations, two unknowns
Newton's law of restitution compares the speed the spheres separate at with the speed they approached at:
The coefficient of restitution e satisfies 0 ≤ e ≤ 1. At e = 1 the impact is perfectly elastic and no kinetic energy is lost; at e = 0 the spheres coalesce. Both speeds are relative speeds along the line of centres, so in terms of velocities the law reads v2 − v1 = e(u1 − u2), and getting that subtraction the right way round is most of the difficulty.
WORKED EXAMPLE
Solving a direct impact
A sphere of mass 2 kg moving at 5 m/s strikes a stationary sphere of mass 3 kg directly. The coefficient of restitution is 0.4. Find both velocities afterwards.
Momentum: 2(5) = 2v1 + 3v2, so 2v1 + 3v2 = 10.
Restitution: v2 − v1 = 0.4(5 − 0) = 2.
Substituting: 2v1 + 3(v1 + 2) = 10 gives 5v1 = 4, so v1 = 0.8 m/s and v2 = 2.8 m/s.
Both are positive, so both spheres move on in the original direction, which is consistent with the heavier one being struck.
Where the energy goes
Momentum is conserved because the internal forces are equal and opposite. Kinetic energy is not, because some of it goes into deforming the spheres and into sound and heat. The opening claim confuses the two: the only collision that keeps all its kinetic energy is the perfectly elastic one with e = 1.
To find the loss, calculate ½mv² for every sphere before and after and subtract the totals. Kinetic energy is a scalar, so signs make no difference to it and squaring removes them anyway, which makes this the one part of the calculation where direction can be ignored.
YOUR TURN
The loss in that impact
For the collision above, find the kinetic energy lost.
Show the working
Before: ½(2)(5²) = 25 J; the stationary sphere contributes nothing.
After: ½(2)(0.8²) + ½(3)(2.8²) = 0.64 + 11.76 = 12.4 J.
Loss = 25 − 12.4 = 12.6 J, over half the original energy.
With e = 1 the velocities would have been −1 and 4, giving 1 + 24 = 25 J and no loss at all.
THE EXAM BIT
- Draw before and after diagrams with a chosen positive direction and label every velocity.
- Write the restitution equation as separation over approach; reversing it gives a negative e.
- Solve the two equations simultaneously rather than guessing which sphere moves where.
- Check the answers make physical sense: the rear sphere cannot end up faster than the one in front.
CHECK YOURSELF
Two spheres approach each other at a combined 8 m/s and separate at 3 m/s. Find the coefficient of restitution.
Show a hint
Separation over approach.
Show the answer
e
=
3
/
8
=
0
.
3
7
5
,
w
h
i
c
h
l
i
e
s
b
e
t
w
e
e
n
0
a
n
d
1
a
s
i
t
m
u
s
t
.
Newton's law of restitution says the separation speed is e times the approach speed, with 0 ≤ e ≤ 1.
Use it with conservation of momentum to get two equations; kinetic energy is lost unless e = 1.
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
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device only.
- State Newton's law of restitution and the range of values e can take.
- Solve a direct collision using conservation of momentum and restitution together.
- Calculate the kinetic energy lost in an impact.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.