A 7.5 kg object moving at 7.3 m/s collides inelastically with a 4.0 kg object which is initially at rest. What percentage of the initial kinetic energy of the system is lost during the collision?
To calculate the percentage of the initial kinetic energy lost during the collision, we need to consider the initial and final kinetic energies of the system.
The initial kinetic energy (Ke) of the system is given by the equation Ke = (1/2)mv^2, where m is the mass and v is the velocity of the object.
For the 7.5 kg object, the initial kinetic energy (Ke1) is:
Ke1 = (1/2) * 7.5 kg * (7.3 m/s)^2 = 0.5 * 7.5 kg * 53.29 m^2/s^2 = 200.18 J
For the 4.0 kg object, since it is initially at rest, its initial kinetic energy (Ke2) is:
Ke2 = 0 J
During an inelastic collision, the two objects combine and move together after the collision. To find the final velocity (vf) of the combined mass, we use the conservation of momentum, which states that the total momentum before the collision is equal to the total momentum after the collision.
Total momentum before collision: (7.5 kg * 7.3 m/s) + (4.0 kg * 0 m/s) = 54.75 kg·m/s
Total momentum after collision: (7.5 kg + 4.0 kg) * vf = 11.5 kg * vf
Setting up the equation:
54.75 kg·m/s = 11.5 kg * vf
Solving for vf:
vf = 54.75 kg·m/s / 11.5 kg
vf ≈ 4.76 m/s
To find the final kinetic energy (Ke_final) after the collision, we use the equation Ke_final = (1/2)mv^2:
Ke_final = (1/2) * 11.5 kg * (4.76 m/s)^2 = 0.5 * 11.5 kg * 22.6576 m^2/s^2 = 129.79 J
The percentage of kinetic energy lost during the collision can be calculated using the formula:
Percentage lost = ((Ke_initial - Ke_final) / Ke_initial) * 100
Substituting the values into the formula:
Percentage lost = ((200.18 J - 129.79 J) / 200.18 J) * 100 ≈ 35.06%
Therefore, approximately 35.06% of the initial kinetic energy of the system is lost during the collision.