A ball is dropped from a height of 10 m. If the energy of the ball reduces by 40% after striking
the ground, how much high can the ball bounce back? (g=10m/s2)
To find the height to which the ball will bounce back, we need to find the reduced energy after striking the ground and then use the conservation of mechanical energy.
Let's start by calculating the reduced energy of the ball after striking the ground. The energy of the ball is given by its potential energy and can be calculated using the formula:
E = mgh
Where:
E = Energy
m = mass of the ball (which is not given in the question but assuming it to be 1 kg for simplicity)
g = acceleration due to gravity (g = 10 m/s^2)
h = height of the ball
Based on the given information, the ball starts at a height of 10 m. Therefore, the initial potential energy is:
E_initial = m * g * h_initial
After striking the ground, the energy of the ball is reduced by 40%. So, the reduced energy is:
E_reduced = 0.6 * E_initial
Now, we can use the conservation of mechanical energy to find the height to which the ball will bounce back. According to the conservation of mechanical energy, the sum of initial potential energy and final kinetic energy should be equal:
E_initial = E_reduced + K
Where:
K = final kinetic energy of the ball
Since the ball strikes the ground, it loses all of its potential energy and gains kinetic energy. The kinetic energy is given by:
K = 0.5 * m * v^2
Where:
v = final velocity of the ball (after bouncing back)
Now, let's solve the equation:
E_initial = E_reduced + K
m * g * h_initial = 0.6 * m * g * h_initial + 0.5 * m * v^2
Canceling out the mass:
g * h_initial = 0.6 * g * h_initial + 0.5 * v^2
Rearranging the equation:
v^2 = (g * h_initial - 0.6 * g * h_initial) / 0.5
v^2 = 0.4 * g * h_initial / 0.5
v^2 = 0.8 * g * h_initial
Now that we have the relation between v^2 and h_initial, we can substitute this in the equation for kinetic energy:
K = 0.5 * m * v^2
K = 0.5 * 1 * 0.8 * g * h_initial
K = 0.4 * g * h_initial
Since the ball bounces back, its final kinetic energy should also be equal to its initial potential energy (E_initial):
K = E_initial
0.4 * g * h_initial = m * g * h_initial
0.4 * g = g
0.4 = 1
Therefore, there's no height to which the ball will bounce back. Due to the energy loss, the ball won't bounce back.