Two objects, m1 = 0.6 kg and m2 = 4.4 kg undergo a one-dimensional head-on collision as shown in thediagram.
Their initial velocities along the one-dimension path are vi1 = 32.4 m/s [right] and vi2 = 8.6 m/s [left].
The two objects stick together after the perfectly inelastic collision
a) Calculate the velocity after the collision.
b) Determine how much kinetic energy is lost due to the collision.
To calculate the velocity after the collision, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision should be equal to the total momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. So, we can calculate the initial momentum before the collision using the formula:
Initial momentum (m1 * vi1) + (m2 * vi2) = Total momentum before collision
Substituting the given values:
(0.6 kg * 32.4 m/s) + (4.4 kg * -8.6 m/s) = Total momentum before collision
Now, we need to find the velocity after the collision. Since the objects stick together after the collision, we can assume that they move with a common final velocity (vf).
The total momentum after the collision is then given by the product of the total mass (m1 + m2) and the final velocity (vf). So, we have:
Total momentum after collision = (m1 + m2) * vf
Therefore, equating the total momentum before the collision to the total momentum after the collision, we get:
(0.6 kg * 32.4 m/s) + (4.4 kg * -8.6 m/s) = (0.6 kg + 4.4 kg) * vf
Simplifying the equation:
(0.6 kg * 32.4 m/s) - (4.4 kg * 8.6 m/s) = (0.6 kg + 4.4 kg) * vf
Finally, we can solve this equation to find the final velocity (vf).
Let's calculate it!
Finding the initial momentum:
(0.6 kg * 32.4 m/s) + (4.4 kg * -8.6 m/s)
= 19.44 kg∙m/s - 37.84 kg∙m/s
= -18.4 kg∙m/s
Calculating the final velocity:
((-18.4 kg∙m/s) / (0.6 kg + 4.4 kg))
= (-18.4 kg∙m/s) / (5 kg)
= -3.68 m/s
Therefore, the velocity after the collision is -3.68 m/s.
Now let's move on to the second part of the question, determining the amount of kinetic energy lost due to the collision.
To calculate the kinetic energy before the collision, we can use the formula:
Initial kinetic energy = (1/2) * m1 * (vi1)^2 + (1/2) * m2 * (vi2)^2
Substituting the given values:
(1/2) * 0.6 kg * (32.4 m/s)^2 + (1/2) * 4.4 kg * (8.6 m/s)^2
Calculating the initial kinetic energy:
= (0.5) * 0.6 kg * 1051.776 m^2/s^2 + (0.5) * 4.4 kg * 73.96 m^2/s^2
= 315.5328 J + 162.544 J
= 478.0768 J
Now, let's calculate the final kinetic energy after the collision. Since the two objects stick together, their final kinetic energy can be calculated using the formula:
Final kinetic energy = (1/2) * (m1 + m2) * (vf)^2
Substituting the given values:
(1/2) * (0.6 kg + 4.4 kg) * (-3.68 m/s)^2
Calculating the final kinetic energy:
= (0.5) * 5 kg * 13.5424 m^2/s^2
= 33.856 J
Finally, the amount of kinetic energy lost due to the collision can be determined by subtracting the final kinetic energy from the initial kinetic energy:
Amount of kinetic energy lost = Initial kinetic energy - Final kinetic energy
= 478.0768 J - 33.856 J
= 444.2208 J
Therefore, the amount of kinetic energy lost due to the collision is 444.2208 Joules.