In the vertical jump, Kobe Bryant starts from a crouch and jumps upward to reach as high as possible. Even the best athletes spend little more than 1.00 second in the air (their "hang time"). Treat Kobe 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 moving up to the time it takes him to go from the floor to that height. You may ignore air resistance.
To calculate the ratio of the time Kobe Bryant is above half of his maximum height (ymax/2) while moving up to the time it takes him to reach that height, we can use the kinematic equations of motion.
Let's denote the time it takes for Kobe Bryant to reach half of his maximum height as t1.
We know that the time it takes for an object to reach a certain height is given by the equation:
t = sqrt(2h/g)
Where:
- t is the time taken
- h is the height
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
In this case, we want to find the time it takes for Kobe Bryant to reach half of his maximum height, so h = ymax/2.
t1 = sqrt(2(ymax/2)/g)
t1 = sqrt(ymax/g)
Next, we need to determine the time Kobe Bryant is above ymax/2 while moving up. Let's denote this time as t2.
Since Kobe spends a little more than 1.00 second in the air, the total time he is above ymax/2 can be calculated as:
t2 = 1 - t1
Finally, the ratio of the time he is above ymax/2 moving up to the time it takes him to go from the floor to that height is:
Ratio = t2 / t1
Substituting the expression for t2 and t1:
Ratio = (1 - t1) / t1
Therefore, to calculate the ratio, you need to first determine the value of t1 using the equation t1 = sqrt(ymax/g), and then substitute it into the expression for the ratio.