A ball of mass 8kg falls from rest from a height of 100m.neglecting air resistance ,calculate it's total energy after falling a distance of 40m
To calculate the ball's total energy after falling a distance of 40m, we need to consider the conservation of mechanical energy. The mechanical energy of an object at any point can be found by summing up its potential energy (PE) and kinetic energy (KE).
The potential energy (PE) of an object at a certain height is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
In this case, the initial potential energy of the ball when it was at a height of 100m would be PE_initial = mgh = 8 kg * 9.8 m/s^2 * 100 m = 7840 J.
The final potential energy of the ball when it is at a height of 40m can be calculated in the same way: PE_final = mgh = 8 kg * 9.8 m/s^2 * 40 m = 3136 J.
Next, we need to consider the change in potential energy as the ball falls. The change in potential energy (ΔPE) is given by the formula ΔPE = PE_final - PE_initial:
ΔPE = 3136 J - 7840 J = -4704 J.
Since the ball is falling, the change in potential energy is negative.
According to the conservation of energy, the change in potential energy is equal to the change in kinetic energy (ΔKE). So, the change in kinetic energy (ΔKE) can be calculated as follows:
ΔKE = -ΔPE = -(-4704 J) = 4704 J.
Finally, we can calculate the final kinetic energy (KE_final) using the formula KE = 1/2 * mv^2, where m is the mass of the object and v is its velocity. Since the ball is falling from rest, its initial velocity is 0.
KE_final = 1/2 * mv^2 = 1/2 * 8 kg * (v^2).
Since we don't have the velocity, we can use the following equation to find it:
v^2 = 2 * ΔKE / m.
Plugging in the values, we have:
v^2 = 2 * 4704 J / 8 kg = 1176 m^2/s^2.
Taking the square root of both sides, we find:
v = √(1176 m^2/s^2) = 34.29 m/s.
Now, we can substitute this value back into the equation for kinetic energy:
KE_final = 1/2 * mv^2 = 1/2 * 8 kg * (34.29 m/s)^2 ≈ 4667.6 J.
Therefore, the total energy of the ball after falling a distance of 40m is approximately 4667.6 J.