A high diver leaves the end of a 5.0 m high diving board and strikes the water 1.4 s later, 2.8 m beyond the end of the board. Considering the diver as a particle, determine the following.
(a) her initial velocity, v0
1 m/s 2° 3
(b) the maximum height above the water reached
4 m
(c) the velocity vf with which she enters the water
To determine the required values for the high diver's motion, we can use the equations of motion. Let's break down the problem step by step:
Step 1: Finding the initial velocity (v0):
We can use the equation of motion that relates displacement (d), initial velocity (v0), time (t), and acceleration (a):
d = v0t + (1/2)at²
In this case, the displacement (d) is the horizontal distance covered by the diver, which is 2.8 m. The time (t) is given as 1.4 s. And since the diver falls vertically, the acceleration (a) can be assumed to be -9.8 m/s² (taking downward as the negative direction).
Substituting the known values into the equation, we have:
2.8 = v0 * 1.4 + (1/2) * (-9.8) * (1.4)²
Simplifying the equation, we can find the value of the initial velocity (v0).
Step 2: Finding the maximum height above the water reached:
To find the maximum height, we can use the equation for vertical motion:
vf² = v0² + 2ad
In this case, the final velocity (vf) is 0 since the diver reaches the highest point where the velocity is zero. The initial velocity (v0) can be determined from the previous step. The acceleration (a) remains -9.8 m/s². And the displacement (d) is the vertical distance covered by the diver, which is equal to the height of the diving board, 5.0 m.
Using this equation, we can find the maximum height (d) above the water reached by substituting the known values:
0 = v0² + 2 * (-9.8) * 5.0
Simplifying the equation, we can solve for the maximum height.
Step 3: Finding the velocity (vf) with which she enters the water:
To find the velocity with which the diver enters the water, we can use the equation of motion:
vf = v0 + at
In this case, the final velocity (vf) can be determined by using the value 0, since the diver comes to rest in the water. The initial velocity (v0) can be obtained from the first step. The acceleration (a) remains -9.8 m/s². And the time (t) for the entire motion is given as 1.4 s.
Using this equation, we can calculate the velocity (vf) by substituting the known values:
0 = v0 + (-9.8) * 1.4
Solving the equation will give us the velocity with which the diver enters the water.
By following these steps, we can determine the values of (a) the initial velocity (v0), (b) the maximum height above the water reached, and (c) the velocity with which she enters the water.