Two railroad freight cars with masses 110 Mg and 170Mg approach with equal speeds of 0.330 . They collide, the lighter car rebounding opposite its original direction at 0.270 m/s.
Find the velocity of the heavier car after the collision. Assume the original direction of the lighter car is positive.
What fraction of the original kinetic energy was lost in this inelastic collision?
Well, this is quite a heavy situation! Let's see what we can do.
To find the velocity of the heavier car after the collision, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
The initial momentum is given by the equation:
(mass of lighter car) x (velocity of lighter car) + (mass of heavier car) x (velocity of heavier car) = (total momentum)
So, plugging in the given values:
(110 Mg) x (0.330 m/s) + (170 Mg) x (0) = (total momentum)
Since the lighter car rebounds, its velocity after the collision is negative:
(110 Mg) x (-0.270 m/s) + (170 Mg) x (velocity of heavier car) = (total momentum)
Now we can solve for the velocity of the heavier car:
(110 Mg) x (-0.270 m/s) + (170 Mg) x (velocity of heavier car) = 0
Simplifying that equation, we get:
-29.7 Mg + 170 Mg x (velocity of heavier car) = 0
Rearranging and solving for the velocity of the heavier car, we find:
(velocity of heavier car) = 29.7 Mg / 170 Mg
And simplifying the fraction, we get:
(velocity of heavier car) = 0.1747 m/s
So, the velocity of the heavier car after the collision is approximately 0.1747 m/s.
Now let's tackle the fraction of the original kinetic energy lost. An inelastic collision means that kinetic energy is not conserved.
The initial kinetic energy before the collision is given by the equation:
(1/2) x (mass of lighter car) x (velocity of lighter car)^2 + (1/2) x (mass of heavier car) x (velocity of heavier car)^2
and the final kinetic energy after the collision is:
(1/2) x (mass of lighter car) x (negative velocity of lighter car)^2 + (1/2) x (mass of heavier car) x (velocity of heavier car)^2
To calculate the fraction of kinetic energy lost, we can use the formula:
(fraction of kinetic energy lost) = (initial kinetic energy - final kinetic energy) / (initial kinetic energy)
Now, let's crunch some numbers and find that fraction!
Remembering that the original direction of the lighter car is positive, we can substitute in the values given and calculate the fraction:
(initial kinetic energy - final kinetic energy) = [(1/2) x (110 Mg) x (0.330 m/s)^2 + (1/2) x (170 Mg) x (0.330 m/s)^2] - [(1/2) x (110 Mg) x (-0.270 m/s)^2 + (1/2) x (170 Mg) x (0.1747 m/s)^2]
(initial kinetic energy - final kinetic energy) = 30.7365 Mg m^2/s^2 - 11.87362 Mg m^2/s^2
(initial kinetic energy - final kinetic energy) = 18.86288 Mg m^2/s^2
Now, divide that by the initial kinetic energy:
(fraction of kinetic energy lost) = (18.86288 Mg m^2/s^2) / [(1/2) x (110 Mg) x (0.330 m/s)^2 + (1/2) x (170 Mg) x (0.330 m/s)^2]
(fraction of kinetic energy lost) = 18.86288 / [0.5 x 110 x (0.330)^2 + 0.5 x 170 x (0.330)^2]
(fraction of kinetic energy lost) = 18.86288 / 11.2713
(fraction of kinetic energy lost) ≈ 1.676
Therefore, approximately 1.676 of the original kinetic energy was lost in this inelastic collision.
Remember, this is all just an approximation since we involved the values in Mg, but the concept is clear.
To solve this problem, we can apply the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Step 1: Determine the momentum of each freight car before the collision.
The momentum of an object is given by the product of its mass and velocity.
Let's denote the velocity of the lighter car as v1 = 0.330 m/s.
The momentum of the lighter car before the collision is given by:
momentum1_before = mass1 * velocity1 = 110 Mg * 0.330 = 36.3 Mg*m/s
For the heavier car, let the velocity be v2 before the collision. Since both cars have equal speeds, the momentum of the heavier car is given by:
momentum2_before = mass2 * velocity2 = 170 Mg * 0.330 = 56.1 Mg*m/s
Step 2: Determine the total momentum before the collision.
The total momentum before the collision is the sum of the individual momenta of the two cars:
total_momentum_before = momentum1_before + momentum2_before
= 36.3 Mg*m/s + 56.1 Mg*m/s
= 92.4 Mg*m/s
Step 3: Determine the momentum of each freight car after the collision.
According to the principle of conservation of momentum, the total momentum after the collision is also equal to the total momentum before the collision.
Let the velocity of the lighter car after the collision be v1' = -0.270 m/s (opposite direction).
Hence, the momentum of the lighter car after the collision is given by:
momentum1_after = mass1 * velocity1' = 110 Mg * (-0.270) = -29.7 Mg*m/s (negative sign indicates opposite direction)
For the heavier car, let the velocity be v2' after the collision.
The momentum of the heavier car after the collision is given by:
momentum2_after = mass2 * velocity2'
Step 4: Determine the velocity of the heavier car after the collision.
Using the principle of conservation of momentum, we can write:
total_momentum_before = total_momentum_after
Therefore,
92.4 Mg*m/s = -29.7 Mg*m/s + momentum2_after
Rearranging the equation, we have:
momentum2_after = 92.4 Mg*m/s + 29.7 Mg*m/s
= 122.1 Mg*m/s
Finally, we can calculate the velocity of the heavier car after the collision:
momentum2_after = mass2 * velocity2'
122.1 Mg*m/s = 170 Mg * velocity2'
velocity2' = 122.1 Mg*m/s / 170 Mg
velocity2' = 0.718 m/s
Therefore, the velocity of the heavier car after the collision is 0.718 m/s.
Step 5: Determine the fraction of the original kinetic energy lost during the collision.
The initial kinetic energy before the collision is given by:
initial_kinetic_energy = (1/2) * mass1 * velocity1^2 + (1/2) * mass2 * velocity2^2
The final kinetic energy after the collision is given by:
final_kinetic_energy = (1/2) * mass1 * velocity1'^2 + (1/2) * mass2 * velocity2'^2
The fraction of kinetic energy lost is given by:
fraction_lost = (initial_kinetic_energy - final_kinetic_energy) / initial_kinetic_energy
Substituting the known values, we can calculate the fraction of kinetic energy lost:
initial_kinetic_energy = (1/2) * 110 Mg * (0.330 m/s)^2 + (1/2) * 170 Mg * (0.330 m/s)^2
= 19.02015 Mg * m^2/s^2 + 29.0477 Mg * m^2/s^2
= 48.06785 Mg * m^2/s^2
final_kinetic_energy = (1/2) * 110 Mg * (-0.270 m/s)^2 + (1/2) * 170 Mg * (0.718 m/s)^2
= 8.28535 Mg * m^2/s^2 + 53.06044 Mg * m^2/s^2
= 61.34579 Mg * m^2/s^2
fraction_lost = (48.06785 Mg * m^2/s^2 - 61.34579 Mg * m^2/s^2) / 48.06785 Mg * m^2/s^2
= -0.27
Therefore, the fraction of the original kinetic energy lost during the collision is -0.27 (negative indicates energy gained).
To find the velocity of the heavier car after the collision, we can use the law of conservation of momentum. According to this law, 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: momentum = mass × velocity.
Let's assign variables to the various quantities given in the problem:
Mass of lighter car (m1) = 110 Mg = 110,000 kg
Mass of heavier car (m2) = 170 Mg = 170,000 kg
Initial velocity of both cars (v) = 0.330 m/s
Velocity of lighter car after collision (v1') = -0.270 m/s (negative sign indicates opposite direction)
The total initial momentum before the collision (P_initial) is given by:
P_initial = m1 × v + m2 × v
The total final momentum after the collision (P_final) is given by:
P_final = m1 × v1' + m2 × v2'
Since momentum is conserved, P_initial = P_final:
m1 × v + m2 × v = m1 × v1' + m2 × v2'
Now, we can rearrange this equation to solve for v2', which is the velocity of the heavier car after the collision:
v2' = (m1 × v + m2 × v - m1 × v1') / m2
Plugging in the given values:
v2' = (110,000 kg × 0.330 m/s + 170,000 kg × 0.330 m/s - 110,000 kg × -0.270 m/s) / 170,000 kg
After calculating this expression, we find that the velocity of the heavier car after the collision is approximately 0.181 m/s.
Now, to determine the fraction of the original kinetic energy lost in this inelastic collision, we can use the formula for kinetic energy:
Kinetic energy = (1/2) × mass × velocity^2
The initial total kinetic energy (KE_initial) is given by:
KE_initial = (1/2) × m1 × v^2 + (1/2) × m2 × v^2
The final total kinetic energy (KE_final) is given by:
KE_final = (1/2) × m1 × v1'^2 + (1/2) × m2 × v2'^2
The fraction of kinetic energy lost in the collision is then:
(Kinetic energy lost) / (Initial kinetic energy) = (KE_initial - KE_final) / KE_initial
Plugging in the given values, we can calculate this fraction.