Two inelastic balls of masses 2 and 3 kg are moving at the velocities 8 and 4 m/s. Find the increase of the internal energy if the first ball catches up the second one.
To find the increase in internal energy when the first ball catches up with the second ball, we can use the principle of conservation of momentum. According to the conservation of momentum, the total momentum before collision is equal to the total momentum after collision.
The total momentum before collision (p_initial) is given by the sum of the individual momenta of the two balls:
p_initial = (mass of first ball) * (velocity of first ball) + (mass of second ball) * (velocity of second ball)
p_initial = (2 kg) * (8 m/s) + (3 kg) * (4 m/s)
To find the total momentum after collision (p_final), we need to consider that the first ball catches up with the second ball. This means that they will be moving together after the collision with a common velocity (let's call it v_final). Since they both have the same velocity after the collision, we can calculate the total momentum by using the sum of their masses multiplied by the common velocity:
p_final = (mass of first ball + mass of second ball) * (v_final)
p_final = (2 kg + 3 kg) * (v_final)
Since the total momentum before and after the collision must be equal, we have:
p_initial = p_final
(2 kg) * (8 m/s) + (3 kg) * (4 m/s) = (2 kg + 3 kg) * (v_final)
Now, we solve the equation for v_final:
16 kg·m/s + 12 kg·m/s = 5 kg * (v_final)
28 kg·m/s = 5 kg * (v_final)
v_final = 28 kg·m/s / 5 kg
v_final = 5.6 m/s
To find the increase in internal energy (ΔU), we can use the equation:
ΔU = (1/2) * (mass of first ball) * (final velocity of the first ball)^2 - (1/2) * (mass of first ball) * (initial velocity of the first ball)^2
ΔU = (1/2) * (2 kg) * (5.6 m/s)^2 - (1/2) * (2 kg) * (8 m/s)^2
Now, we can solve for ΔU:
ΔU = 0.5 * 2 kg * (5.6 m/s)^2 - 0.5 * 2 kg * (8 m/s)^2
ΔU = 0.5 * 2 kg * (31.36 m^2/s^2) - 0.5 * 2 kg * (64 m^2/s^2)
ΔU = 31.36 J - 64 J
ΔU = -32.64 J
Therefore, the increase in internal energy is -32.64 Joules. Note that the negative sign indicates a decrease in internal energy, which signifies that some energy was lost during the collision.