A ball is thrown from ground at an angle θ with horizontal and with an initial speed u₀. For the resulting projectile motion, the magnitude of average velocity of the ball up to the point when it hits the ground for the first time is V₁. After hitting the ground, ball rebounds at the same angle θ but with a reduced speed of u₀/α. Its motion continues for a long time as shown in figure. If the magnitude of average velocity of the ball for entire duration of motion is 0.8 V₁, the value of α is_______.

Question

G1 Physics · Accepted Answer

For successive bounces, speeds are u₀, u₀/α, u₀/α², … . For a fixed rebound angle, each range is proportional to speed² and each flight time to speed. Thus total range = R₁[1 + α⁻² + α⁻⁴ + …] and total time = T₁[1 + α⁻¹ + α⁻² + …]. Since V₁ = R₁/T₁, the overall average speed is V₁[(1 - α⁻¹)/(1 - α⁻²)] = V₁·α/(α + 1). Given this is 0.8V₁, α/(α + 1) = 0.8, so α = 4.