Pushing apart
Step from a canoe onto a dock and the canoe shoots backward, sometimes leaving you with one foot on the dock and one in the lake. Two skaters at rest push off each other and glide apart. A fireworks shell bursts into glowing fragments. These are all explosions (or recoil) in the physics sense: one object, or a group of objects, pushes apart because of forces inside the system.
Explosions are collisions run backward. The same law, conservation of momentum, governs both.
Explosions from rest
If a system starts at rest, its total momentum is zero. The forces that push the pieces apart are internal, so the total momentum is still zero afterwards. For two pieces:
0 = m₁v₁′ + m₂v₂′, so m₁v₁′ = −m₂v₂′
- The pieces move in opposite directions.
- Their momenta are equal in size.
- Their speeds are inversely proportional to their masses: v₁′ ÷ v₂′ = m₂ ÷ m₁ (in size). The lighter piece moves faster.
Where does the kinetic energy come from?
Before the explosion, the kinetic energy is zero. Afterwards, both pieces are moving, so the kinetic energy has increased. This energy comes from energy stored inside the system: elastic potential energy in a compressed spring (from the sibling unit Work, Energy and Simple Harmonic Motion, Ee = ½kx²), chemical energy in gunpowder or fuel, or chemical energy in your muscles.
Momentum is conserved in an explosion. Kinetic energy is not: it increases. Total energy is still conserved, because the stored energy decreases by the same amount (some of it also becomes sound and thermal energy).
Since Ek = p² ÷ (2m) and the two momenta are equal in size, the lighter piece always gets more kinetic energy: Ek₁ ÷ Ek₂ = m₂ ÷ m₁.
Explosions of moving objects
If the system is already moving when it breaks apart, the total momentum after is equal to the (non-zero) total momentum before:
(m₁ + m₂)v = m₁v₁′ + m₂v₂′
This describes a fireworks shell bursting in flight, a rocket dropping a spent stage, or a person jumping off a moving toboggan.
Hands-on activityOffice-chair explosions
You need: two rolling office chairs, a partner, a smooth hard floor with lots of open space, a tape measure, and a bathroom scale. Grown-up note: clear the area, keep feet up so they do not drag, and push gently.
- Weigh each person. Sit in the chairs facing each other, knees almost touching, chairs at rest.
- Put your palms together and push apart in one smooth push. Do not touch the floor.
- Measure how far each chair rolls before stopping. Repeat three times and average.
- Repeat with a heavy backpack added to one person's lap (weigh the backpack too).
- Compare the ratio of distances with the ratio of masses (each person plus their chair; if you cannot weigh the chairs, assume they are equal).
What to look for: the lighter side rolls farther. Right after the push, the speeds are in the inverse ratio of the masses (v_light ÷ v_heavy = m_heavy ÷ m_light). If rolling friction slows both chairs in the same way, the distance rolled is proportional to the starting speed squared, so you should find roughly d_light ÷ d_heavy ≈ (m_heavy ÷ m_light)². Your data will not match perfectly, because chair wheels differ. Either way, the total momentum right after the push is zero.
Check your understandingQuiz
1. A 1.0 kg cart and a 3.0 kg cart are pushed apart from rest by a spring. The 3.0 kg cart moves at 1.0 m/s. What is the speed of the 1.0 kg cart?
2. A system at rest explodes into several pieces. Which statement is true?
3. Two pieces fly apart from rest. Which piece has more kinetic energy?
4. When you step out of a canoe toward a dock, the canoe moves away from the dock because:
5. A shell moving at 10 m/s [E] bursts into two pieces. The total momentum of the pieces just after the burst is:
6. A 0.60 kg object at rest explodes into two pieces. A 0.20 kg piece moves at 6.0 m/s. What is the speed of the other piece?