A 9000 kg boxcar traveling at 17.0 m/s strikes a second boxcar at rest. The two stick together and move off with a speed of 6.5 m/s. What is the mass of the second car?
If the two cars stuck together, it is an inelastic collision, and energy is not conserved during the collision.
However, momentum is conserved, and since there is only one common final velocity, v, only the momentum equation is enough to solve for the unknown mass.
The equation of conservation of momentum:
m1 v1 + m2 v2 = (m1+m2)v ..... (1)
m1=9000 kg
m2=unknown
v1=17 m/s
v2=0
v=6.5 m/s
Solve for m2.
To find the mass of the second car, we can use the principle of conservation of momentum.
According to the principle of conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
The momentum before the collision is given by:
Momentum before = (mass of first car) * (velocity of first car) + (mass of second car) * (velocity of second car)
The momentum after the collision is given by:
Momentum after = (mass of first car + mass of second car) * (velocity after the collision)
Since the first car strikes the second car at rest, the velocity of the second car before the collision is 0 m/s.
Therefore, we have:
Momentum before = (mass of first car) * (velocity of first car) + (mass of second car) * (0)
Momentum after = (mass of first car + mass of second car) * (velocity after the collision)
According to the conservation of momentum, the two equations are equal:
(mass of first car) * (velocity of first car) = (mass of first car + mass of second car) * (velocity after the collision)
Substituting the given values:
(9000 kg) * (17.0 m/s) = (9000 kg + mass of second car) * (6.5 m/s)
Simplifying the equation:
153000 kg·m/s = 58500 kg·m/s + (mass of second car) * (6.5 m/s)
Rearranging the equation:
58500 kg·m/s = (mass of second car) * (6.5 m/s)
Dividing both sides of the equation by 6.5 m/s:
(mass of second car) = 58500 kg·m/s / 6.5 m/s
Simplifying:
(mass of second car) = 9000 kg
So, the mass of the second car is 9000 kg.
To find the mass of the second car, we can use 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.
The momentum of an object is given by the product of its mass and velocity. So, we can write the equation for momentum conservation as:
(mass of first car) * (velocity of first car) + (mass of second car) * 0 = (mass of first car + mass of second car) * (velocity of combined cars)
Given:
Mass of first car (m1) = 9000 kg
Velocity of first car (v1) = 17.0 m/s
Velocity of combined cars (v_combined) = 6.5 m/s
We need to find the mass of the second car (m2).
Using the momentum conservation equation, we can rewrite it as an equation to solve for m2:
(m1 * v1) + (m2 * 0) = (m1 + m2) * v_combined
Substituting the known values:
(9000 kg * 17.0 m/s) + (m2 * 0) = (9000 kg + m2) * 6.5 m/s
Simplying the equation:
(153,000 kg * m/s) = (9000 kg + m2) * 6.5 m/s
Solving for m2:
153,000 kg * m/s = 5,850 kg * m/s + 6.5 m/s * m2
Now, let's isolate the m2 term by moving the other terms to the other side of the equation:
153,000 kg * m/s - 5,850 kg * m/s = 6.5 m/s * m2
Simplifying further:
147,150 kg * m/s = 6.5 m/s * m2
Now, divide both sides by 6.5 m/s:
m2 = 147,150 kg * m/s / 6.5 m/s
m2 ≈ 22,629.2307 kg (rounded to four decimal places)
Therefore, the mass of the second car is approximately 22,629.2307 kg.