A cannon on a railroad car is facing in a direction parallel to the tracks. It fires a 99 kg shell at a speed of 105 m/s (relative to the ground) at an angle of 60.0° above the horizontal. If the cannon plus car have a mass of 5.5 104 kg, what is the recoil speed of the car if it was at rest before the cannon was fired?
To find the recoil speed of the car, we can apply the law of conservation of momentum. According to this law, the total momentum before the cannon is fired should be equal to the total momentum after the cannon is fired.
The momentum of an object is its mass multiplied by its velocity. In this case, the momentum of the shell is given as follows:
Momentum of shell = mass of shell × velocity of shell
Given: mass of shell = 99 kg, velocity of shell = 105 m/s
Momentum of shell = 99 kg × 105 m/s = 10,395 kg·m/s
To calculate the initial momentum of the cannon plus car, we need to find the initial velocity of the cannon plus car. Since it is at rest before the cannon is fired, the initial velocity is 0 m/s.
Momentum of cannon plus car (initial) = (mass of cannon plus car) × (initial velocity of cannon plus car)
Given: mass of cannon plus car = 5.5 × 10^4 kg, initial velocity of cannon plus car = 0 m/s
Momentum of cannon plus car (initial) = 5.5 × 10^4 kg × 0 m/s = 0 kg·m/s
According to the law of conservation of momentum, the total momentum before and after the cannon is fired should be the same. Therefore:
Momentum of shell = Momentum of cannon plus car (final)
10,395 kg·m/s = (mass of cannon plus car) × (final velocity of cannon plus car)
We can rearrange the equation to solve for the final velocity of the cannon plus car:
Final velocity of cannon plus car = (Momentum of shell) / (mass of cannon plus car)
Final velocity of cannon plus car = 10,395 kg·m/s / 5.5 × 10^4 kg
Final velocity of cannon plus car ≈ 0.189 m/s (rounded to three decimal places)
Therefore, the recoil speed of the car, when the cannon is fired, is approximately 0.189 m/s in the direction opposite to the firing of the shell.