A ball of mass 1kg is dropped from a height of 7m and rebounds to a height of 4.5m. Calculate:
(I) its KE just before impact
(II) its initial rebound velocity and KE
(III) Account for the loss of KE on the impact.
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To solve this problem, we need to consider the conservation of energy. We will use the formulas for gravitational potential energy (GPE) and kinetic energy (KE). Let's calculate each part step by step:
(I) To find the kinetic energy just before impact, we need to calculate the potential energy at a height of 7m and subtract it from the potential energy at a height of 4.5m.
GPE at 7m = mass * gravity * height = 1kg * 9.8 m/s^2 * 7m = 68.6 J
GPE at 4.5m = mass * gravity * height = 1kg * 9.8 m/s^2 * 4.5m = 44.1 J
The kinetic energy just before impact can be found by subtracting the potential energy at 4.5m from the potential energy at 7m:
KE = GPE at 7m - GPE at 4.5m = 68.6 J - 44.1 J = 24.5 J
So, the kinetic energy just before impact is 24.5 Joules.
(II) To find the initial rebound velocity and kinetic energy, we need to consider the conservation of mechanical energy. At the highest point of the rebound (4.5m), the ball has only potential energy, which will be converted into kinetic energy at the maximum height.
Using the formula for potential energy, we can find the initial potential energy at 4.5m:
GPE at 4.5m = mass * gravity * height = 1kg * 9.8 m/s^2 * 4.5m = 44.1 J
Since the ball reaches its maximum height and comes to rest, the kinetic energy at the highest point is zero. Therefore, we can equate the initial potential energy to the final kinetic energy:
KE = GPE at 4.5m
KE = 44.1 J
Now, we can calculate the initial rebound velocity using the formula for kinetic energy:
KE = (1/2) * mass * velocity^2
Rearranging the formula, we get:
velocity^2 = (2 * KE) / mass
velocity^2 = (2 * 44.1 J) / 1kg
velocity^2 = 88.2 m^2/s^2
Finally, taking the square root of both sides, we can find the initial rebound velocity:
velocity = sqrt(88.2 m^2/s^2)
velocity ≈ 9.4 m/s
Therefore, the initial rebound velocity is approximately 9.4 m/s, and the initial kinetic energy is 44.1 Joules.
(III) To account for the loss of kinetic energy on impact, we can compare the initial kinetic energy (44.1 J) with the kinetic energy just before impact (24.5 J). The difference represents the loss of kinetic energy:
Loss of KE = Initial KE - Final KE
Loss of KE = 44.1 J – 24.5 J
Loss of KE = 19.6 J
Therefore, the loss of kinetic energy on impact is 19.6 Joules.