A 1.30 × 10
4
kg railroad car moving at 6.70
m/s to the north collides with and sticks to
another railroad car of the same mass that is
moving in the same direction at 1.83 m/s.
What is the velocity of the joined cars after
the collision?
Answer in units of m/
To find the velocity of the joined cars after the collision, 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 formula for momentum is given by: momentum = mass × velocity
Given:
Mass of each railroad car = 1.30 × 10^4 kg
Initial velocity of the first car (moving to the north) = 6.70 m/s
Initial velocity of the second car (moving in the same direction) = 1.83 m/s
Let's assume the final velocity of the joined cars after the collision is v.
Using the conservation of momentum, we can write the equation as:
(mass of first car × initial velocity of first car) + (mass of second car × initial velocity of second car) = (mass of first car + mass of second car) × final velocity of joined cars
(1.30 × 10^4 kg × 6.70 m/s) + (1.30 × 10^4 kg × 1.83 m/s) = (2 × 1.30 × 10^4 kg) × v
Simplifying the equation:
(8.71 × 10^4 kg m/s) + (2.379 × 10^4 kg m/s) = (2.60 × 10^4 kg) × v
(11.089 × 10^4 kg m/s) = (2.60 × 10^4 kg) × v
Dividing both sides of the equation by (2.60 × 10^4 kg):
11.089 × 10^4 kg m/s / (2.60 × 10^4 kg) = v
4.265 m/s = v
Therefore, the velocity of the joined cars after the collision is 4.265 m/s.
To find the velocity of the joined cars after the collision, 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 defined as the product of its mass and velocity. Therefore, we can calculate the initial momentum of the first car and the second car, and then use the conservation of momentum to find the velocity after the collision.
Let's calculate the initial momentum of each car:
Initial momentum of the first car = mass of first car × velocity of first car
= (1.30 × 10^4 kg) × (6.70 m/s) = 8.71 × 10^4 kg·m/s
Initial momentum of the second car = mass of second car × velocity of second car
= (1.30 × 10^4 kg) × (1.83 m/s) = 2.379 × 10^4 kg·m/s
The total initial momentum before the collision is the sum of the individual momenta:
Total initial momentum = Initial momentum of first car + Initial momentum of second car
= 8.71 × 10^4 kg·m/s + 2.379 × 10^4 kg·m/s = 1.015 × 10^5 kg·m/s
According to the law of conservation of momentum, the total momentum after the collision is equal to the total momentum before the collision. Since the cars stick together after the collision, their combined mass is twice the mass of an individual car.
Let's assume the velocity of the joined cars after the collision is v, then the total momentum after the collision is:
Total momentum after collision = Total mass of joined cars × Velocity of joined cars
= (2 × 1.30 × 10^4 kg) × v = 2.60 × 10^4 kg·m/s · v
Since the total momentum before and after the collision is the same, we can set up an equation:
Total momentum before collision = Total momentum after collision
1.015 × 10^5 kg·m/s = 2.60 × 10^4 kg·m/s · v
Now, we can solve for v:
v = (1.015 × 10^5 kg·m/s) / (2.60 × 10^4 kg·m/s)
v ≈ 3.90 m/s
Therefore, the velocity of the joined cars after the collision is approximately 3.90 m/s to the north.