Bouncing Ball

Scenario

A marble, dropped from rest at height H above the floor, is allowed to bounce off the floor two times. The collision with the floor is completely elastic and negligibly short. Sketch the marbles’s v-t, s-t and a-t graphs.

Solution

  • (xmtutor)
  • (xmdemo)
  • Since this motion involves both upward and downward velocity, the more intuitive “upward is positive” sign convention is adopted.

2 vt.gif

  • The marble starts from rest, accelerates at a constant rate of 9.81 m s-2 before colliding with the floor at the maximum speed of v0.
  • Switching direction abruptly, the marble rebounds with speed v0.
  • Again moving under the influence of gravity only, the marble decelerates at 9.81 m s-2, returning to rest at the starting position.

2 st.gif

  • The s-t graph consists of quadratic curve segments between bounces.

2 at.gif

  • The marble has a constant downward acceleration of g except when it is in contact with the floor.
  • During each bounce, the velocity changes from –v0 to v0 in a negligibly short duration of time. This abrupt change in velocity is a very large acceleration. (Think a=\frac{\Delta v}{\Delta t}=\frac{2{{v}_{0}}}{\Delta t}, where Δt is the very short duration of contact.) This explains the spikes on the a-t graphs.

 

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