Question

In the vertical jump, an athlete starts from a crouch and jumps upward as high as possible. Even the best athletes spend little more than 1.00 s in the air (their "hang time"). Treat the athlete as a particle and let ymax be his maximum height above the floor. To explain why he seems to hang in the air, calculate the ratio of the time he is above ymax/2 to the time it takes him to go from the floor to that height. Ignore air resistance.

Answer 1

Answer:

t_up / t_down = 6.83

Explanation:

Find:

Calculate the ratio of the time he is above y_max/2 to the time it takes him to go from the floor to that height.

Solution:

- Compute the velocity v_o at y_max:

                             v_i^2 = v_f^2 - 2*g*y_max

                             0 = v_o^2 - 2*g*y_max

                             v_o = sqrt (2*g*y_max)

- The total time spend by athlete above height y_max / 2 is:

                             y - y_o = v_o*t_up - 0.5*g*t^2_up

                             v_o = 0.5*g*t_up

- Equate two equations:

                             sqrt (2*g*y_max) = 0.5*g*t_up

                             t_up = 2*sqrt(2*g*y_max) / g

- The total time taken by athlete to reach height y_max / 2 from ground is:

                            y - y_o = v_o*t_down - 0.5*g*t^2_down

                            g*t_^2down - 2*v_o*t_down + y_max = 0

- Solve the quadratic and evaluate t_down:

                            t_down = (v_o +/- sqrt (v^2_o - g*y_max)) / g

Substitute for v_o =  sqrt (2*g*y_max)

                            t_down = (sqrt(2g*y_max) +/- sqrt(g*y_max)) / g

- We will use the minus quantity, because we need the first part of the journey from ground from the two times he passes the height of y_max/2.

Hence,

                             t_down = (sqrt(2g*y_max) - sqrt(g*y_max)) / g

                             t_down = (sqrt(g*y_max) / g) * (sqrt(2) - 1)

- Compute the ratio t_up to t_down:

         t_up / t_down = 2*sqrt(2*g*y_max) / g * g / (sqrt(g*y_max)*(sqrt(2) - 1)

                                  = 2*sqrt(2) / (sqrt(2) - 1)

                                  = 6.83