A boy reaches out of a window and tosses a ball straight up with a speed of 15 m/s. The ball is 20 m above the ground as he releases it. Use energy to find the following.
The following what? How high it goes above the ground? Call that height Ymax.
g*H + Vo^2/2 = g*Ymax
H = 20 m
Vo = 15 m/s
g = 9.81 m/s^2
Solve for Ymax
To find the following values using energy, we can analyze the initial and final states of the ball.
1. Maximum height reached by the ball:
When the ball reaches its maximum height, its speed will momentarily be zero. At this point, all of its initial kinetic energy will have been converted to gravitational potential energy. Therefore, we can equate the initial kinetic energy to the final potential energy to find the maximum height.
The initial kinetic energy (KEi) can be calculated using the formula:
KEi = (1/2) * mass * (speed)^2
The final potential energy (PEf) can be calculated using the formula:
PEf = mass * gravity * height
So, setting KEi equal to PEf, we have:
(1/2) * mass * (speed)^2 = mass * gravity * height
We can simplify this equation to find the maximum height (height_max):
height_max = (speed)^2 / (2 * gravity)
Substituting the given values into this equation will give us the answer.
2. Time taken for the ball to reach its maximum height:
We can use the formula for vertical motion to find the time taken for the ball to reach its maximum height. The formula for displacement in vertical motion is:
displacement = initial velocity * time + (1/2) * acceleration * (time)^2
In this case, the initial velocity is the upward speed of the ball (15 m/s), the displacement is the maximum height (which we found in the previous step), and the acceleration is the acceleration due to gravity (-9.8 m/s^2). Solving this equation for time will give us the answer.
3. Time taken for the ball to fall back to the boy's hand:
The time taken for the ball to fall back to the boy's hand is the same as the time taken for it to reach its maximum height. This is because the motion of the ball is symmetrical in time.
By using these equations, we can find the maximum height, time taken to reach maximum height, and time taken to fall back to the boy's hand using the given values of speed (15 m/s) and height (20 m).