Mass 1 with a mass of 28.5 kg starts at rest at the top of a frictionless from a height of 17 meters. It then slides down the incline and moves over a horizontal frictionless surface and collides inelastically (head on) with mass 2 which is of unknown mass. After the collision, the total kinetic energy of both masses is 60% of the initial total kinetic energy of both masses before the collision. What is the amount of mass 2 in kg?
To solve this problem, we can use the law of conservation of energy. The initial potential energy is converted into the sum of the final kinetic energy and internal energy (due to the collision).
1. Calculate the initial potential energy of mass 1:
Potential energy = mass * gravity * height
Potential energy = 28.5 kg * 9.8 m/s^2 * 17 m
2. Calculate the initial kinetic energy of mass 1:
Initial total kinetic energy = Initial kinetic energy of mass 1
Initial total kinetic energy = Potential energy
3. Calculate the final total kinetic energy of both masses:
Final total kinetic energy = 0.6 * Initial total kinetic energy
4. Determine the initial kinetic energy of mass 2:
Initial kinetic energy of mass 2 = Final total kinetic energy - Initial kinetic energy of mass 1
5. We know that kinetic energy is given by the formula:
Kinetic energy = (1/2) * mass * velocity^2
Since both masses collide head-on in an inelastic collision, they have the same final velocity after the collision.
6. Write the equation for the initial kinetic energy of mass 2:
Initial kinetic energy of mass 2 = (1/2) * mass 2 * velocity^2
7. Equate the initial kinetic energy of mass 2 to the value obtained in step 4:
(1/2) * mass 2 * velocity^2 = Initial kinetic energy of mass 2
8. Solve for mass 2:
mass 2 = (2 * Initial kinetic energy of mass 2) / velocity^2
By following these steps, you can find the amount of mass 2 in kg.
To solve this problem, we can use the principle of conservation of mechanical energy, specifically the conservation of kinetic energy. The conservation of kinetic energy states that the total kinetic energy of a system remains constant, assuming there are no external forces acting on the system.
1. Let's start by calculating the initial kinetic energy of Mass 1 just before it starts sliding down the incline. We can use the formula for kinetic energy: KE = 0.5 * mass * velocity^2.
Since Mass 1 is at rest, its initial velocity is 0. Therefore, the initial kinetic energy of Mass 1 is 0.
2. Next, let's calculate the potential energy of Mass 1 when it is at the top of the incline. The formula for potential energy is PE = mass * gravity * height.
PE = 28.5 kg * 9.8 m/s^2 * 17 m = 4767.3 J (Joules)
This potential energy will be converted into kinetic energy as Mass 1 slides down the incline.
3. Now, let's calculate the final kinetic energy of both masses after the collision. We know that it is 60% of the initial total kinetic energy. Since Mass 1 was at rest initially, the total initial kinetic energy is equal to the final kinetic energy of Mass 2.
Let's assume that Mass 2 has a mass of M2 kg. The final kinetic energy of Mass 2 is 60% of the initial total kinetic energy, which is 0.6 times the sum of the initial kinetic energy of Mass 1 (0 J) and the potential energy converted to kinetic energy (4767.3 J) of Mass 1.
Final kinetic energy of Mass 2 = 0.6 * (0 J + 4767.3 J) = 0.6 * 4767.3 J = 2860.38 J
4. Now, we can use the formula for kinetic energy to find the final velocity of Mass 2. Rearranging the formula, we get: velocity = sqrt(2 * KE / mass).
Plugging in the values, we have:
2860.38 J = 0.5 * M2 kg * velocity^2
velocity^2 = (2860.38 J) / (0.5 * M2 kg)
velocity = sqrt((2860.38 J) / (0.5 * M2 kg))
5. Since we have calculated the final velocity of Mass 2, we can use the formula momentum = mass * velocity to find the momentum of Mass 2.
Momentum of Mass 2 = M2 kg * velocity
6. Finally, we need to realize that since the collision is inelastic, the momentum is conserved before and after the collision. Therefore, the momentum of Mass 2 after the collision is equal to the momentum of Mass 1 before the collision.
The momentum of Mass 1 before the collision is 0 since it was at rest.
Equating the momentum of Mass 2 after the collision to 0, we have:
Momentum of Mass 2 = 0
M2 kg * velocity = 0
M2 kg = 0
Therefore, the mass of Mass 2 is 0 kg.
Therefore, the amount of Mass 2 is 0 kg.