An oblique collision occurs between two objects with a similar rectangular shape (1m x 0.2m) as shown below. The first object has a mass of 10 kg and is initially traveling at 4 m/s. The second object has a mass of 6 kg and is initially at rest. The impact point, P, is 0.4 m from the center of mass of each object. The coefficient of restitution for this impact is 0.5. Determine the percentage of the kinetic energy lost from the impact in %. (Answer is between 0 and 100)
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To determine the percentage of kinetic energy lost from the impact, we need to compare the initial total kinetic energy of the system to the final total kinetic energy after the collision.
The initial total kinetic energy can be calculated by finding the individual kinetic energy of each object and summing them up.
For the first object, the initial kinetic energy is given by:
KE1 = (1/2) * mass1 * velocity1^2
Substituting the given values:
KE1 = (1/2) * 10 kg * (4 m/s)^2
KE1 = 80 J
For the second object, since it is initially at rest, its initial kinetic energy is zero.
Therefore, the initial total kinetic energy of the system is 80 J.
Now, let's consider the final total kinetic energy. To do that, we need to determine the final velocities of the two objects after the collision.
Since this is an oblique collision, we need to use the principle of conservation of linear momentum for both the x and y directions separately.
In the x-direction, the initial momentum is given by:
Px = mass1 * velocity1
Substituting the given values:
Px = 10 kg * 4 m/s
Px = 40 kg·m/s
Since momentum is conserved, the final momentum in the x-direction will also be 40 kg·m/s.
Similarly, in the y-direction, the initial momentum is zero because both objects are moving in the x-direction.
Now, let's consider the vertical component of the collision. Since the objects have similar rectangular shapes, the collision is assumed to be elastic in the y-direction.
The final momentum in the y-direction will also be zero because the objects will bounce back symmetrically in opposite directions.
Considering the initial and final momenta in both directions, we can now calculate the final velocities.
In the x-direction, the final velocities will be the same for both objects because they have the same mass and collide at the same distance from their centers of mass (impact point P).
Using the coefficient of restitution (e), we can write the equation for the relative velocity after impact in the x-direction:
relative velocity in x-direction = (e * (velocity2 - velocity1)) / (1 + (mass1/mass2))
Substituting the given values:
(velocity2 - 4 m/s) = (0.5 * (velocity2 - 4 m/s)) / (1 + (10 kg/6 kg))
Simplifying, we can solve for velocity2:
(velocity2 - 4 m/s) = 0.4167 * (velocity2 - 4 m/s)
velocity2 - 4 m/s = 0.4167 * velocity2 - 1.6667 m/s
0.5833 * velocity2 = 2.3333 m/s
velocity2 = 4 m/s (approximately)
Therefore, after the collision, both objects will have a final velocity of 4 m/s in the x-direction.
Now, let's calculate the final total kinetic energy:
For the first object, the final kinetic energy is given by:
KE1_final = (1/2) * mass1 * velocity1_final^2
Substituting the given values:
KE1_final = (1/2) * 10 kg * (4 m/s)^2
KE1_final = 80 J
For the second object, the final kinetic energy is given by:
KE2_final = (1/2) * mass2 * velocity2_final^2
Substituting the given values:
KE2_final = (1/2) * 6 kg * (4 m/s)^2
KE2_final = 48 J
Therefore, the final total kinetic energy of the system is 80 J + 48 J = 128 J.
Now, to determine the percentage of kinetic energy lost from the impact, we can calculate the difference between the initial and final total kinetic energy:
Kinetic energy lost = Initial total kinetic energy - Final total kinetic energy
Kinetic energy lost = 80 J - 128 J
Kinetic energy lost = -48 J
Since the kinetic energy cannot be negative, we take the absolute value:
Kinetic energy lost = |-48 J| = 48 J
To calculate the percentage of kinetic energy lost, we need to compare the lost energy to the initial total kinetic energy:
Percentage kinetic energy lost = (Kinetic energy lost / Initial total kinetic energy) * 100
Percentage kinetic energy lost = (48 J / 80 J) * 100
Percentage kinetic energy lost ≈ 60%
Therefore, the percentage of kinetic energy lost from the impact is approximately 60%.