Two carts: rectilinear collision and Newton's second law, part I

Post date: Jan 31, 2014 7:42:32 AM

Two carts are moving along a level track, that is, their motion is confined in one dimension. When the two carts collide, each cart undergoes a change in momentum, i.e. a change in velocity. Each cart is acted upon by a net force during the interval of collision or interaction. Assume that the direction to the right is positive. Assume furthermore that friction is negligible. The figure shows the horizontal forces acting on each cart during the interval of collision.

Let us consider cart B. The net force exerted by cart A on cart B is  . This net force varies in time, and the possible plot is shown. 

If we apply Newton's second law,    and integrate both sides over the time interval, we obtain:

The area under the force-versus-time graph (or the integral) has a name, ``impulse delivered by the force".  Moreover, it is equal to the change in momentum of cart B.

If we measure the velocities before and after collision and calculate the momentum change, we will know the impulse. Furthermore, we will know the history of the force even if we do not know the variation of force instant by instant. 

The above figures are found in Arnold B. Arons, Teaching Introductory Physics (John Wiley & Sons, Inc., USA, 1997).