A railroad car with a mass of 12400 kg collides and couples with a second car of mass 18700 kg that is initially at rest. The first car is moving with a speed of 7.7 m/s prior to the collision.
(a) What is the initial momentum of the first car?
(b) If external forces can be ignored, what is the final velocity of the two railroad cars after they couple?
(a) The initial momentum of the first car can be calculated by multiplying its mass (12400 kg) with its velocity (7.7 m/s).
Initial momentum = Mass × Velocity
= 12400 kg × 7.7 m/s
So, the initial momentum of the first car is 95,480 kg·m/s.
(b) Since we are assuming that no external forces are acting on the system, the law of conservation of momentum can be applied. According to this law, the total momentum before the collision is equal to the total momentum after the collision.
Total momentum before collision = Total momentum after collision
The total momentum before the collision is equal to the initial momentum of the first car, which is 95,480 kg·m/s.
Let's assume the final velocity of the coupled cars is v.
The momentum of the second car, initially at rest, is zero.
Total momentum after collision = (Mass of first car + Mass of second car) × Final velocity
= (12400 kg + 18700 kg) × v
Therefore, we can set up an equation:
Total momentum before collision = Total momentum after collision
95,480 kg·m/s = (31,100 kg) × v
To solve for v, we can divide both sides of the equation by 31,100 kg:
v = 95,480 kg·m/s / 31,100 kg
By simplifying the equation, we find that the final velocity of the two coupled railroad cars is approximately 3.07 m/s.
So, the final velocity of the coupled railroad cars after the collision is approximately 3.07 m/s.
To solve this problem, 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, assuming no external forces act on the system.
(a) To calculate the initial momentum of the first car, we can use the formula:
Momentum = mass * velocity
Given:
Mass of the first car, m1 = 12400 kg
Initial velocity of the first car, v1 = 7.7 m/s
Using the formula:
Initial momentum of the first car = m1 * v1
= 12400 kg * 7.7 m/s
= 95,480 kg·m/s
Therefore, the initial momentum of the first car is 95,480 kg·m/s.
(b) To calculate the final velocity of the two railroad cars after they couple, we can continue using the principle of conservation of momentum.
The initial momentum of the second car is zero since it is initially at rest. Therefore, the total initial momentum of the system is equal to the initial momentum of the first car only.
Using the formula:
Total initial momentum = Initial momentum of the first car
Total initial momentum = m1 * v1
Total initial momentum = 12400 kg * 7.7 m/s
Now, since the two cars are coupled after the collision, they move together as one object. To calculate the final velocity after they couple, we can use the formula:
Total final momentum = Total initial momentum
Total final momentum = (m1 + m2) * vf
where:
m2 = mass of the second car
vf = final velocity of the two cars after they couple
Given:
Mass of the second car, m2 = 18700 kg
Using the formula:
Total final momentum = (m1 + m2) * vf
Total final momentum = (12400 kg + 18700 kg) * vf
The total final momentum is equal to the initial momentum of the first car:
(12400 kg + 18700 kg) * vf = 12400 kg * 7.7 m/s
Calculating further:
31100 kg * vf = 12400 kg * 7.7 m/s
Dividing both sides by 31100 kg:
vf = (12400 kg * 7.7 m/s) / 31100 kg
vf ≈ 3.092 m/s
Therefore, the final velocity of the two railroad cars after they couple is approximately 3.092 m/s.
To solve this problem, we need to 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.
(a) To find the initial momentum of the first car, we can use the formula:
Momentum = mass × velocity
So, the initial momentum of the first car (before the collision) is given by:
Momentum₁ = mass₁ × velocity₁
Here, the mass of the first car is 12400 kg, and the velocity is 7.7 m/s. Plugging in the values:
Momentum₁ = 12400 kg × 7.7 m/s
Now you can calculate the initial momentum of the first car.
(b) To find the final velocity of the coupled cars, we can use the principle of conservation of momentum as mentioned earlier.
According to the principle of conservation of momentum:
Momentum₁ + Momentum₂ = Total Momentum
Initially, the second car is at rest, which means its initial momentum is zero. So, the equation becomes:
Momentum₁ + 0 = Total Momentum
The total momentum after the cars couple is equal to the total mass of the two cars times their final velocity:
Total Momentum = (mass₁ + mass₂) × velocity
Let's represent the final velocity of the coupled cars as V. Now we can write the equation as:
Momentum₁ + 0 = (mass₁ + mass₂) × V
Substituting the values:
12400 kg × 7.7 m/s + 0 = (12400 kg + 18700 kg) × V
Now, you can solve this equation to find the final velocity (V) of the two cars after they couple.
a. Momentum=m1*V1 = 12,400 * 7.7=95,480
b. Momentum = (m1+m2)*V = 95,480
m1 = 12,400 kg
m2 = 18,700 kg
Solve for V.