Question

6. For this series of 4 questions, a 1-1b ball is dropped from rest and falls a distance before striking the smooth plane at A. It bounces from the plane, and then strikes the plane again. You will be asked to solve for different things. Please pay attention: the numbers may change from problem to problem since they are randomized. 2) If the ball is dropped form rest at a height h = 4.5 ft, and the coefficient of restitution between the ball and the plane is e = 0.7, what is its speed right after it strikes the plane at A? (Hint: when striking the wall/ground, the line of impact is always normal to the surface, and the wall/ground always has a speed of zero.) Your answer must include 2 places after the decimal point. Take g = 32.2 ft/s2.

Answer 1

By energy conservation, speed just before the impact is given as;

$\frac{1}{2}mv_o^2 = mgh$

$\Rightarrow v_o = \sqrt{2gh} = \sqrt{2\times32.2\times4.5} = 17.0235\;\;ft/s$

Component of speed before impact, parallel to the slope;

$v_{o,\parallel }=\frac{3}{5}v_o$

$\Rightarrow v_{o,\parallel }=\frac{3}{5}(17.0235)$

$\Rightarrow v_{o,\parallel }=10.2141\;\;ft/s$

Component of speed before impact, perpendicular to the slope;

$v_{o,\perp }=\frac{4}{5}v_o$

$\Rightarrow v_{o,\perp }=\frac{4}{5}(17.0235)$

$\Rightarrow v_{o,\perp }=13.6188\;\;ft/s$

Component of speed after impact, perpendicular to the slope;

$v_{\perp }=ev_{o,\perp }=0.7(13.6188\;\;ft/s)$

$\Rightarrow v_{\perp }=9.5332\;\;ft/s$

Velocity parallel to the plane will remain the same, hence;

$v_{\parallel }=v_{o,\parallel }=10.2141\;\;ft/s$

Total velocity right after the ball strikes the plane at A is;

$v = \sqrt{9.5332^2+10.2141^2}$

$\Rightarrow v = 13.97\;\;ft/s$