Explosions, Recoil and Momentum

In an explosion, objects that start at rest push apart. The total momentum before is zero, so the total momentum after must also be zero.

Explosions from rest

$0 = m_1 v_1 + m_2 v_2$

$m_1 v_1 = -m_2 v_2$

The two objects move in opposite directions with momenta of equal magnitude.

Example: A 1500 kg cannon fires a 5 kg shell at 300 m/s. Find the recoil velocity.

$0 = 5(300) + 1500(v)$

$v = -\frac{1500}{1500} = -1.0 \text{ m/s}$

The cannon recoils at 1.0 m/s in the opposite direction.

Why the lighter object moves faster

Since the momenta are equal in magnitude: $m_1 v_1 = m_2 v_2$

The lighter object ($m_1 < m_2$) must have the greater speed ($v_1 > v_2$).

Rocket propulsion

A rocket expels exhaust gases at high speed in one direction. By conservation of momentum, the rocket accelerates in the opposite direction.

As fuel is burned, the rocket’s mass decreases, so the same thrust produces a greater acceleration over time.

Common errors and how to correct them

• Forgetting that total momentum before the explosion is zero (objects were at rest).
• Getting confused about direction signs. Pick one direction as positive, stick with it.

How to apply this in an exam

State that total momentum before = 0. Write the momentum equation with appropriate signs. Solve for the unknown velocity.