Momentum is mass × velocity, and impulse is the change in momentum caused by a force acting for a time. In a collision with no outside force, the total momentum stays the same.
This lesson is part of SPM Physics force and motion I. It uses the velocity ideas from using equations of uniformly accelerated motion.
How do I set up a collision?
Choose a positive direction first, then fill in a before-and-after table.
| Mass (kg) | Velocity before (m s⁻¹) | Velocity after (m s⁻¹) | |
|---|---|---|---|
| Car A | 1200 | +5 | v |
| Car B | 800 | 0 | v |
Cars A and B stick together after the collision, so both move at the same velocity v. Total momentum before = 1200 × 5 + 800 × 0 = 6000 kg m s⁻¹. Total momentum after = (1200 + 800)v = 2000v.
Setting them equal, 2000v = 6000, so v = 3.0 m s⁻¹. The table makes it clear that the direction of A was chosen as positive.
Is kinetic energy conserved too?
No. Kinetic energy before = ½ × 1200 × 5² = 15 000 J. After = ½ × 2000 × 3² = 9000 J. The 6000 J that is lost went into sound, heat and deformation of the cars.
This type of collision is called inelastic. Momentum is conserved in every collision without an outside force, but kinetic energy is conserved only in elastic ones.
Worked example: impulse and impact time
A 0.15 kg ball travelling at 12 m s⁻¹ is caught and brought to rest in 0.020 s. Find the average force on the hands.
Change in momentum = 0.15 × 0 − 0.15 × 12 = −1.8 kg m s⁻¹, so the magnitude is 1.8 kg m s⁻¹. Force = impulse ÷ time = 1.8 ÷ 0.020 = 90 N.
Now let the hands move back and lengthen the stopping time to 0.10 s. The change in momentum is the same, so the force = 1.8 ÷ 0.10 = 18 N. A five-times longer stopping time gives a five-times smaller force.
The mistake to avoid
The common mistake is to drop the direction when the ball bounces back. A ball of mass 0.15 kg hits a wall at +10 m s⁻¹ and returns at −8 m s⁻¹.
| Method | Change in momentum |
|---|---|
| Wrong: 0.15 × (8 − 10) | −0.3 kg m s⁻¹ |
| Right: 0.15 × (−8 − 10) | −2.7 kg m s⁻¹ |
The wrong version treats the return speed as positive. When the direction reverses, the two speeds add, so a bounce gives a larger impulse than a stop.
Check yourself
A 2.0 kg trolley moving at 3.0 m s⁻¹ hits a stationary 1.0 kg trolley, and the two stick together. Find their common velocity and the kinetic energy lost.
Answer
Momentum before = 2.0 × 3.0 = 6.0 kg m s⁻¹. After: 3.0v = 6.0, so v = 2.0 m s⁻¹.
Kinetic energy before = ½ × 2.0 × 9.0 = 9.0 J. After = ½ × 3.0 × 4.0 = 6.0 J. Energy lost = 3.0 J.
What to study next
Go on to drawing force diagrams, which shows how forces cause the changes in momentum you have just calculated. Record sign mistakes in the mistake log and paper-error review tool, and try the force and motion I practice set.
If you want a teacher to go through your collision tables with you, see online one-to-one Physics tuition.