You work for NASA designing a low-cost landing system for a Mars mission. The payload will be surrounded by padding and dropping onto the surface. When it reaches thesurface, it will bounce. The height and the distance of the bounces will get smaller with each bounce so that it finally comes to rest on the surface. Your boss asksyou to determine how the ratio of the horizontal distance covered by two successive bounces depends on the ratio of the heights of the two bounces and the ratio of thehorizontal components of the initial velocity of the two bounces. After making the calculation you decide to check it in your laboratory on Earth.
a) Draw a sketch of the situation, including velocity and acceleration vectors at all relevant times! During what time interval does the ball have motion that iseasiest to calculate? Is the acceleration of the ball during that time interval constant or is it changing? Are the time durations of two successive bounces equal?Label the horizontal distances and maximum heights for each of the first two bounces.
b) Write down the basic kinematics equations that apply to the time intervals you selected, under the assumptions you have made.
c) Write and equation for the horizontal distance the ball travels in the air during the first bounce, in terms of the initial horizontal velocity of the ball, itshorizontal acceleration, and the time it stays in the air before reaching the ground again.
d) The equation you just wrote contains the time of flight, which must be re-written in terms of other quantities. Determine it from the vertical motion of the ball.First, select an equation that gives the ball's vertical position during a bounce as a function of its initial vertical velocity, its vertical acceleration, and thetime elapsed since it last touched the ground.
e) How's the ball vertical position when it touches the ground at the "end" of its first bounce related to its vertical position when it touched the ground at the"beginning" of its first bounce? Use this relationship and the equation from step 4 to write one/two equation involving the time of flight.
f) Combine the previous steps to get an equation for the horizontal distance of a bounce in terms of the ball's horizontal velocity, the height of the bounce, and theball's vertical acceleration.
g) Repeat the above process for the next bounce; take the ratio of horizontal distances to get your prediction equation.