A ball falls freely a height onto and smooth inclined plane forming an angle α with the horizontal. If the ratio of the distance between the points at which the jumping ball strikes the inclined plane is 1:2:x, find the value of x. Assume the impacts to be elastic.
Let x1x2x3 be the point of collision of three consecutive jump ball approaches with velocity v from point B in every jump time taken is same say (+).
x1=(vcosα)t+12gsinαt2→(1)
at point B velocity in x-direction is
v′=(vcosα+gsinαt)
x2=v′t+12gsinαt2=(vcosα+gsinαt)t+12gsinαt2→(ii)
given that xx2=12
Therefore gsint2=vcosαt+12gsinαt2
at point of c velocity in x-direction will be
gsint2=vcosαt+12gsinαt2v′′=(vcosα+2gsinαt)x3=v"t+12gsinαt2=(vcosα+2gsinαt)t+12gsinαt2x3=3(vcosα+12gsinαt2)→(iii)
from equation (1) and (3)
x1x3=13
therefore ratio x:x2:x3=1:2:3