- Michael Jordan’s vertical leap is reported to be 48 inches. What ishis takeoff speed? Give your answer in meters per second.
- After a 65-newton weight has fallen freely from rest a vertical distance of 5.3 meters, the kinetic energy of the weight is 1
a. 12 J
b. 340 J
c. 910 J
d. 1800 J
The Correct Answer and Explanation is :
1. Michael Jordan’s Takeoff Speed
To calculate Michael Jordan’s takeoff speed based on his vertical leap, we can use the principle of conservation of mechanical energy. At the peak of his jump, his kinetic energy (speed) is completely converted to potential energy. The formula for potential energy is:
[
PE = mgh
]
where:
- (PE) is the potential energy,
- (m) is the mass,
- (g) is the acceleration due to gravity (approximately 9.81 m/s²),
- (h) is the height (in meters).
From the information given, the height (h) is 48 inches, which is about 1.22 meters.
When he leaves the ground, his speed converts all of his kinetic energy into this potential energy, so:
[
KE = PE
]
This implies:
[
\frac{1}{2}mv^2 = mgh
]
where:
- (KE) is kinetic energy,
- (v) is the speed (takeoff speed),
- (m) is mass.
We can solve for (v), the takeoff speed:
[
\frac{1}{2}v^2 = gh
]
[
v^2 = 2gh
]
[
v = \sqrt{2gh}
]
Substituting the values:
[
v = \sqrt{2 \times 9.81 \times 1.22}
]
[
v = \sqrt{23.92} \approx 4.89 \, \text{m/s}
]
Thus, Michael Jordan’s takeoff speed is approximately 4.89 meters per second.
2. Kinetic Energy of the Weight After Falling
Given:
- The weight is 65 newtons,
- It falls a distance of 5.3 meters.
First, let’s calculate the mass of the weight. The force of gravity acting on an object is (F = mg), where:
- (F) is the force (65 N),
- (m) is the mass,
- (g) is the acceleration due to gravity (9.81 m/s²).
Rearranging for mass:
[
m = \frac{F}{g} = \frac{65}{9.81} \approx 6.63 \, \text{kg}
]
Now, we can calculate the velocity of the weight just before it hits the ground using the kinematic equation:
[
v^2 = u^2 + 2gh
]
where:
- (v) is the final velocity (just before hitting the ground),
- (u) is the initial velocity (0 m/s, since it falls from rest),
- (g) is the acceleration due to gravity,
- (h) is the height it falls (5.3 meters).
Substituting the known values:
[
v^2 = 0 + 2 \times 9.81 \times 5.3
]
[
v^2 = 103.5
]
[
v = \sqrt{103.5} \approx 10.17 \, \text{m/s}
]
Next, we calculate the kinetic energy (KE) using the formula:
[
KE = \frac{1}{2}mv^2
]
Substituting (m = 6.63 \, \text{kg}) and (v = 10.17 \, \text{m/s}):
[
KE = \frac{1}{2} \times 6.63 \times (10.17)^2
]
[
KE \approx \frac{1}{2} \times 6.63 \times 103.5 \approx 342.6 \, \text{J}
]
The closest answer is 340 J, so the correct answer is:
b. 340 J
Explanation:
The process involves understanding how energy is transferred. For the first problem, Michael Jordan’s vertical leap involves converting kinetic energy (speed) into potential energy as he rises. This conversion process is what allows us to calculate his takeoff speed.
In the second problem, we are given the weight’s force and height fallen. Using the principles of gravitational potential energy and kinetic energy, we can find the mass and speed of the object. From there, applying the kinetic energy formula gives us the energy after falling a certain distance.