A 2.0-kg object moving 5.0 m/s collides with and sticks to an 8.0-kg object initially at rest. Determine the kinetic energy lost by the system as a result of this collision.
To determine the kinetic energy lost by the system during the collision, we need to calculate the initial kinetic energy of the system before the collision and the final kinetic energy of the system after the collision.
1. Calculate the initial kinetic energy (KE) of the system:
- The initial kinetic energy is given by the formula: KE = (1/2) * mass * velocity^2.
- For the 2.0-kg object: KE1 = (1/2) * 2.0 kg * (5.0 m/s)^2.
- Substituting the values: KE1 = 0.5 * 2.0 kg * 25.0 m^2/s^2 = 25.0 J (joules).
2. Calculate the final velocity of the combined object:
- Since the two objects stick together after the collision, their velocities will be the same.
- To find the final velocity (Vf) of the combined object, we can use the principle of conservation of momentum:
Momentum before collision = Momentum after collision.
(mass1 * velocity1) + (mass2 * velocity2) = (mass1 + mass2) * Vf.
- Substituting the values: (2.0 kg * 5.0 m/s) + (8.0 kg * 0 m/s) = (2.0 kg + 8.0 kg) * Vf.
10.0 kg*m/s = 10.0 kg * Vf.
- Solving for Vf: Vf = 10.0 kg*m/s / 10.0 kg = 1.0 m/s.
3. Calculate the final kinetic energy (KEf) of the system:
- The final kinetic energy of the combined object is given by the same formula: KEf = (1/2) * mass * velocity^2.
- For the combined object with a mass of 2.0 kg + 8.0 kg = 10.0 kg and a velocity of 1.0 m/s: KEf = (1/2) * 10.0 kg * (1.0 m/s)^2.
- Substituting the values: KEf = 0.5 * 10.0 kg * 1.0 m^2/s^2 = 5.0 J (joules).
4. Calculate the kinetic energy lost by the system:
- The kinetic energy lost is the difference between the initial kinetic energy (KE1) and the final kinetic energy (KEf).
- KE lost = KE1 - KEf = 25.0 J - 5.0 J = 20.0 J (joules).
Therefore, the kinetic energy lost by the system as a result of this collision is 20.0 joules.
To find the kinetic energy lost by the system, we need to evaluate the initial kinetic energy of the system before the collision and the final kinetic energy of the system after the collision.
The initial kinetic energy is given by the formula:
KE_initial = (1/2) * mass1 * velocity1^2 + (1/2) * mass2 * velocity2^2
where mass1 and mass2 are the masses of the objects, and velocity1 and velocity2 are their respective velocities.
In this case, the initial kinetic energy is:
KE_initial = (1/2) * (2.0 kg) * (5.0 m/s)^2 + (1/2) * (8.0 kg) * (0 m/s)^2
= (1/2) * 2.0 kg * 25.0 m^2/s^2 + 0 J
= 25.0 J
Since the two objects stick together after the collision, their final velocity will be the same. Let's call this final velocity v_final. Using the principle of conservation of momentum, we can find v_final:
(mass1 * velocity1) + (mass2 * velocity2) = (mass1 + mass2) * v_final
(2.0 kg * 5.0 m/s) + (8.0 kg * 0 m/s) = (2.0 kg + 8.0 kg) * v_final
10.0 kg * 5.0 m/s = 10.0 kg * v_final
50.0 kg·m/s = 10.0 kg·v_final
v_final = 5.0 m/s
Now, we can calculate the final kinetic energy of the system, KE_final:
KE_final = (1/2) * (mass1 + mass2) * v_final^2
= (1/2) * (2.0 kg + 8.0 kg) * (5.0 m/s)^2
= (1/2) * 10.0 kg * 25.0 m^2/s^2
= 125.0 J
The kinetic energy lost by the system as a result of this collision is the difference between the initial and final kinetic energies:
KE_lost = KE_initial - KE_final
= 25.0 J - 125.0 J
= -100.0 J
Therefore, the system lost 100.0 J of kinetic energy during the collision.