6.6 Conservation of Mechanical Energy
Racing Balls


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There is a distinction between asking about the final speed and asking about the time of travel.  As explained in Section 6.6 when discussing amusement park rides, the only factor affecting the final speed is the change in height. Since the vertical displacement is the same, we know that both balls will have the same final speed.
However, to find the time of travel we will focus on the horizontal component of the velocity. The displacement along the horizontal dH is related to the average velocity along the horizontal vH and time t by:  dH = vH t
The ball with the larger average horizontal velocity arrives first. As shown in the velocity time graph made by the applet, the first ball accelerates for a short time then moves at a constant velocity. Except for a small region, the horizontal component of the velocity of the second ball is similar to the first. However, in the region of the path drawn in red, ball 2 accelerates beyond the maximum value reached by ball 1 then slows down to reach the final speed.  Since the second ball never moves slower than the first, we can deduce that its average velocity along the horizontal is greater. Thus, it arrives first.
Fu-Kwun Hwang