Question
A 10 g bullet moving at embeds itself in a 990 g wooden block initially at rest on a frictionless horizontal surface. The block, with the bullet inside, then compresses a spring of stiffness . Find the speed of the block + bullet just after collision and the maximum compression of the spring.
Solution — Step by Step
The collision is perfectly inelastic — they stick together. Momentum is conserved (external forces during the brief collision are negligible):
Initial KE of bullet: .
KE just after embedding: .
The other 792 J becomes heat, sound, and deformation — that is the inelastic-collision signature.
After embedding, no friction acts. So all 8 J of KE converts to spring PE at maximum compression:
Final: post-collision speed = 4 m/s, spring compression cm.
Why This Works
Two different conservation laws govern two different stages. During the bullet-block collision, only momentum is conserved because internal forces are huge but brief. Between the collision and the spring stop, there is no friction so mechanical energy is conserved.
Mixing them up — for instance, applying energy conservation through the collision — gives wildly wrong answers. This is the most-tested subtlety in collision problems.
Alternative Method
Skip the intermediate speed by combining steps. Initial momentum gives . Then :
Common Mistake
Students apply , treating the collision as elastic. That gives m, ten times too large. Always check whether a collision is elastic or inelastic before deciding which conservation law to apply.