A basketball player, standing near the basket to grab a rebound, jumps 73.6 cm vertically. How much time does the player spend in the bottom 15.4 cm of the jump?

I need someone to check my work.

I got about 3.797 as the inital velocity. I plugged this into "x = vt + 1/2at^2" and used the quadratic formula to get 0.81391 for t.

Now...do I have to times this number by 2 to get the answer? If not, what exactly do I do?

To find the time the basketball player spends in the bottom 15.4 cm of the jump, you don't need to multiply the time by 2. You have already calculated the time accurately.

To explain step by step:

1. Given the vertical jump height of 73.6 cm, you correctly used the equation x = vt + (1/2)at^2, where:
- x is the displacement (73.6 cm)
- v is the initial velocity (which you calculated as 3.797)
- a is the acceleration (which can be taken as the acceleration due to gravity, -9.8 m/s^2)
- t is the time.

2. You substituted the values into the equation and solved for t.

3. However, when doing the calculations, make sure to use consistent units. Since you used centimeters for displacement and velocity, let's convert acceleration to centimeters as well.

- 1 m = 100 cm
- Thus, -9.8 m/s^2 = -980 cm/s^2.

4. After substituting the values into the equation x = vt + (1/2)at^2, you obtained the quadratic equation:

73.6 = 3.797t + (1/2)(-980)t^2

5. Solving this equation using the quadratic formula, you correctly obtained t ≈ 0.81391 seconds.

6. Finally, this time represents the total time spent in achieving the entire vertical jump. To find the time spent in the bottom 15.4 cm, you can directly use the value of t ≈ 0.81391 seconds since this represents the total time.

Therefore, the basketball player spends approximately 0.81391 seconds in the bottom 15.4 cm of the jump.

To find the amount of time the basketball player spends in the bottom 15.4 cm of the jump, we can start by determining the time it takes for the player to reach the highest point of the jump. We can use the equation of motion:

y = v_i * t + (1/2) * a * t^2

Where:
y = vertical displacement (height)
v_i = initial velocity (upward velocity)
a = acceleration (which is -9.8 m/s^2 for freefall near the Earth's surface)
t = time

In this case, the initial velocity is the upward velocity the basketball player jumps with, and since the question is given in centimeters and seconds, we will use those units.

Now, you mentioned that you calculated the initial velocity to be approximately 3.797 cm/s, which is correct.

So, using the equation with the known values of initial velocity (v_i = 3.797 cm/s), acceleration (a = -9.8 cm/s^2), and the unknown of time (t), we have:

73.6 cm = 3.797 cm/s * t + (1/2) * (-9.8 cm/s^2) * t^2

Simplifying and rearranging the equation:

0.49 cm/s^2 * t^2 + 3.797 cm/s * t - 73.6 cm = 0

Now, to solve this equation and find the time when the player reaches the highest point, we can use the quadratic formula:

t = (-b ± sqrt(b^2 - 4ac)) / (2a)

Applying this formula with the values of a = 0.49 cm/s^2, b = 3.797 cm/s, and c = -73.6 cm, we get:

t = (-3.797 ± sqrt((3.797)^2 - 4 * 0.49 * -73.6)) / (2 * 0.49)

Simplifying further,

t ≈ (-3.797 ± sqrt(14.416057)) / 0.98

Now, you mentioned that you obtained approximately t = 0.81391 seconds, which is correct.

To find the time spent in the bottom 15.4 cm, we can multiply this time by 2, since the player will spend an equal amount of time at the bottom as well as the top. Hence,

Time spent in the bottom 15.4 cm = 2 * 0.81391 seconds

Therefore, the basketball player will spend approximately 1.62782 seconds in the bottom 15.4 cm of the jump.

I hope this clarifies the process for finding the answer and confirms your calculations.