A ball weighing .2 Lbs. is dropped out the window of a pickup truck (approx. 5 ft?) moving horizontally at 45mph. If e (coefficient of restitution) between the ball and the ground is equal to .6, find the height of the ball's first bounce and the horizontal distance it will travel before bouncing again?
To find the height of the ball's first bounce (H1), we need to consider the energy transformation during the collision with the ground. Using the coefficient of restitution (e), which represents the ratio of the final velocity to the initial velocity, we can apply the following formula:
e = sqrt(H1 / H0)
Here, H0 represents the initial drop height. Since the ball is dropped from a height of 5 ft (which is approximately 1.524 meters), H0 = 1.524 meters.
Solving for H1, we can rearrange the formula:
H1 = H0 * e^2
H1 = 1.524 * (.6)^2
H1 ≈ 0.547 meters (rounded to three decimal places)
Hence, the height of the ball's first bounce is approximately 0.547 meters.
Now, to find the horizontal distance the ball will travel before bouncing again, we need to calculate the time it takes for the ball to hit the ground (T1) based on its initial vertical velocity (Vy0) and the acceleration due to gravity (g).
Using the formula:
Vy = Vy0 - g * t
Let's find Vy0 first:
Vy0 = 0 (since the ball is initially dropped with no vertical velocity)
Next, let's find T1:
0 = Vy0 - g * T1
0 = -9.8 * T1
T1 = 0
This result implies that the ball will hit the ground instantaneously.
Now, to find the horizontal distance traveled before the second bounce, we need to calculate the ball's horizontal velocity (Vx) using the formula:
Vx = Vx0
where Vx0 is the initial horizontal velocity of the ball, which is equal to the speed of the pickup truck (45 mph).
Converting 45 mph to meters per second:
Vx0 = 45 * (0.44704 m/s per 1 mph)
= 20.1168 m/s
Since the ball bounces instantaneously and Vx remains constant, we can calculate the horizontal distance (D) traveled before the second bounce using:
D = Vx * T1
D = 20.1168 * 0
D = 0
Hence, the horizontal distance the ball will travel before bouncing again is 0 meters.