A ball is dropped from a height of 5.67 m. If the ball rebounds with a speed of 4.22 m/s, what is the coefficient of restitution?
a. .40
b. .74
c. .56
d. 1.23
I've ruled d out because it is greater than 1.
The coefficient of restitution is the ratio of the velocities just after and just before bouncing.
If it is dropped from a height of H = 5.67 m, its speed when it hits the ground will be sqrt(2*g*H) = 10.54 m/s
The coefficient of restitution is therefore
r = 4.22/10.54 = 0.400
To find the coefficient of restitution, we need to use the formula:
Coefficient of restitution = (speed of rebound)/(speed of impact)
In this case, the speed of rebound is given as 4.22 m/s, but we need to find the speed of impact.
The speed of impact can be calculated using the principle of conservation of mechanical energy. Since the ball is dropped from a height of 5.67 m, we can calculate the initial potential energy and equate it to the kinetic energy at impact (when it reaches the ground level).
Potential energy at initial position = m * g * h
where m is the mass of the ball, g is the gravitational acceleration (9.8 m/s^2), and h is the height (5.67 m).
Kinetic energy at impact = (1/2) * m * v^2
where v is the speed of impact.
Since the potential energy is completely converted to kinetic energy, we can set the two equations equal to each other:
m * g * h = (1/2) * m * v^2
Simplifying, we find:
v = sqrt(2 * g * h)
Substituting the given values, we have:
v = sqrt(2 * 9.8 * 5.67)
v ≈ 11.92 m/s
Now that we have the speed of impact (11.92 m/s) and the speed of rebound (4.22 m/s), we can calculate the coefficient of restitution:
Coefficient of restitution = (speed of rebound)/(speed of impact)
Coefficient of restitution = 4.22/11.92 ≈ 0.354
None of the options provided match this result exactly, so we need to consider rounding errors. Given that option a. 0.40 is the closest value to our result, it would be the most appropriate answer.