Use the equations below:
P=mv
Ek=1/2mv^2
3) A railway carriage of 5.00 x 10^4 moves along a track at 2.50ms-1. It collides with a second, stationary, carriage with a mass of 4.00 X 10^4 kg and the carriages join together.
A) Calculate the initial velocity of the coupled carriages after the impact.
B) Calculate the change in kinetic energy during the collision and hence determine if the collision is elastic or inelastic and why.
10 marks
A. M1*V1 + M2*V2 = M1V + M2V.
5*10^4 * 2.50 + M2*0 = 5*10^4V+4*10^4V,
7.5*10^4 + 0 = 9*10^4V,
V = 0.833 m/s.
B. Before collision: KE1 = 0.5*M1*V1^2.
KE1 = 0.5*5*10^4 * (2.50)^2 = 3.125*10^4 Joules.
After collision: KE2 = 0.5*M1*V^2 + 0.5*M2*V^2,
KE2 = 0.5*5*10^4*V^2 + 0.5*4*10^4*V^2,
KE2 = 4.5*10^4V^2 = 4.5*10^4*(0.833)^2 = 3.123*10^4 Joules.
KE1 = KE2, Kinetic energy conserved!
Therefore we have an elastic collision.
To answer these questions, we'll need to apply the principles of conservation of momentum and conservation of kinetic energy.
A) To calculate the initial velocity of the coupled carriages after the impact, we'll use the principle of conservation of momentum. This principle states that the total momentum before the collision should be equal to the total momentum after the collision.
The formula for momentum is:
P = mv
Let's assume that the initial velocity of the first carriage is v1 and the initial velocity of the second carriage is 0 (since it is stationary). After the collision, the two carriages will move together at a final velocity v2.
Using the conservation of momentum, we can write the equation:
(m1 * v1) + (m2 * 0) = (m1 + m2) * v2
Substituting the values given in the question:
(5.00 * 10^4 kg * 2.50 m/s) + (4.00 * 10^4 kg * 0) = (5.00 * 10^4 kg + 4.00 * 10^4 kg) * v2
Simplifying:
(1.25 * 10^5 kg * m/s) = (9.00 * 10^4 kg) * v2
Dividing both sides by (9.00 * 10^4 kg):
v2 = (1.25 * 10^5 kg * m/s) / (9.00 * 10^4 kg)
v2 = 1.39 m/s (rounded to two decimal places)
Therefore, the initial velocity of the coupled carriages after the impact is 1.39 m/s.
B) To determine if the collision is elastic or inelastic, we need to calculate the change in kinetic energy during the collision.
The formula for kinetic energy is:
Ek = 1/2mv^2
Before the collision, the kinetic energy of the system is given by:
Ek1 = 1/2 * m1 * v1^2 + 1/2 * m2 * 0^2
Ek1 = 1/2 * (5.00 * 10^4 kg) * (2.50 m/s)^2 + 0
After the collision, the kinetic energy of the system is given by:
Ek2 = 1/2 * (m1 + m2) * v2^2
Ek2 = 1/2 * (5.00 * 10^4 kg + 4.00 * 10^4 kg) * (1.39 m/s)^2
Calculating the values:
Ek1 = 1/2 * (5.00 * 10^4 kg) * (2.50 m/s)^2 = 156.25 * 10^3 J
Ek2 = 1/2 * (5.00 * 10^4 kg + 4.00 * 10^4 kg) * (1.39 m/s)^2 = 62.21 * 10^3 J
The change in kinetic energy is given by:
ΔEk = Ek2 - Ek1
ΔEk = (62.21 * 10^3 J) - (156.25 * 10^3 J)
ΔEk = -94.04 * 10^3 J
Since the change in kinetic energy is negative, it means that the collision is inelastic. In an inelastic collision, some kinetic energy is lost due to deformation, sound, or heat. In this case, the negative value indicates a decrease in kinetic energy.
Therefore, the collision between the carriages is inelastic because the change in kinetic energy is negative.