Michael Jordan has the record for the highest vertical jump at 48 inches (1.22m). While on the ground, he speeds up over a distance of 0.6 m to achieve the necessary launch speed. What acceleration does his legs provide in order to achieve his height? (Hint: you will need to split the problem between when he is on the ground and in the air.)
To find the acceleration that Michael Jordan's legs provide in order to achieve his jump height, we can split the problem into two parts: when he is on the ground and when he is in the air.
Step 1: Calculate the initial velocity on the ground.
We know that Michael Jordan speeds up over a distance of 0.6 m to achieve the necessary launch speed. Let's assume that he starts from rest, so his initial velocity on the ground is 0 m/s.
Step 2: Calculate the time it takes for him to reach launch speed.
We can use the equation of motion:
v = u + at
Where:
v = final velocity (launch speed)
u = initial velocity (0 m/s)
a = acceleration
t = time
Rearranging the equation to solve for time (t), we get:
t = (v - u) / a
Since his initial velocity (u) is 0 m/s, the equation simplifies to:
t = v / a
Substituting the values, with v = launch speed (which we don't know yet) and a = acceleration, we get:
t = v / a
Step 3: Calculate the time taken to reach the peak height.
Once Michael Jordan has reached the launch speed, he will be in the air and only under the influence of gravity. The time taken to reach the peak height is the same as the time taken to vertically come to rest after reaching the launch speed.
Using the equation:
v = u + at
Where:
v = final velocity (0 m/s at the peak)
u = initial velocity (launch speed)
a = acceleration due to gravity (approximately 9.8 m/s²)
t = time
Rearranging the equation to solve for time (t), we get:
t = (v - u) / a
Since the final velocity (v) is 0 m/s at the peak, the equation becomes:
t = -u / a
Step 4: Calculate the launch speed.
To determine the launch speed, we need to calculate the time taken to reach the peak height first. We can use the equation derived in Step 2:
t = v / a
Rearranging the equation to solve for velocity (v), we get:
v = a * t
Now we can substitute the value of time (t) from Step 3 (since it is the same as the time taken to reach the peak height) and acceleration due to gravity (a ≈ 9.8 m/s²), and solve for velocity (v):
v = 9.8 * t
Step 5: Calculate the acceleration.
Now we have the launch speed (v) and the distance traveled on the ground (0.6 m). We can use the equation of motion:
v² = u² + 2as
Where:
v = final velocity (launch speed)
u = initial velocity (0 m/s)
a = acceleration (which we want to find)
s = distance traveled (0.6 m)
Rearranging the equation to solve for acceleration (a), we get:
a = (v² - u²) / (2s)
Substituting the values, with v = launch speed, u = 0 m/s, and s = distance on the ground (0.6 m), we can calculate the acceleration (a):
a = (v² - 0) / (2 * 0.6)
Therefore, the acceleration that Michael Jordan's legs provide in order to achieve his jump height can be calculated using the steps mentioned above.