A ball is dropped from a height of 8m onto a horizontal floor and rebounds to a height of 12m.what is the coefficient of restitution between the ball and the floor?take g=10m/s^2
unless it has rocket propulsion it will not drop from 8 and rebound to 12
To find the coefficient of restitution (e), we need to use the formula:
e = (v2f - v1f) / (v1i - v2i)
Where:
- v1i is the initial velocity of the ball before impact
- v2i is the initial velocity of the ball after impact
- v1f is the final velocity of the ball before impact
- v2f is the final velocity of the ball after impact
Let's break down each step to find the values needed for the formula:
Step 1: Find the initial velocity of the ball before impact (v1i).
Since the ball is dropped, it falls freely under gravity. We can find the initial velocity using the equation:
v1i = sqrt(2gh)
Where:
- g is the acceleration due to gravity = 10 m/s^2 (given)
- h is the initial height = 8 m (given)
Plugging in the values, we get:
v1i = sqrt(2 * 10 * 8) = sqrt(160) = 12.65 m/s (approx.)
Step 2: Find the final velocity of the ball before impact (v1f).
Since the ball rebounds, we need to find the final velocity just before it hits the ground. We can use the equation:
v1f^2 = v1i^2 + 2gh
Using the same g and h values as before, we can calculate v1f:
v1f^2 = 12.65^2 + 2 * 10 * 8
v1f^2 = 160 + 160
v1f^2 = 320
v1f = sqrt(320) = 17.89 m/s (approx.)
Step 3: Find the initial velocity of the ball after impact (v2i).
Since the ball rebounds, we know that the initial velocity after impact is negative (opposite direction). Therefore, v2i = -v1f = -(-17.89) = 17.89 m/s
Step 4: Find the final velocity of the ball after impact (v2f).
The final velocity after impact is given as 12 m/s (given).
Now, we can substitute these values into the coefficient of restitution formula:
e = (v2f - v1f) / (v1i - v2i)
e = (12 - 17.89) / (12.65 - (-17.89))
e = (-5.89) / (30.54)
e ≈ -0.192 (approx.)
Therefore, the coefficient of restitution between the ball and the floor is approximately -0.192.