A railway flat car is rushing along a level frictionless track at a speed of 45m/s. Mounted on the car and aimed forward is a cannon that fires 65 kg cannon balls with a muzzle speed of 625m/s. The total mass of the car, the cannon, and the large supply of cannon balls on the car is 3500 kg. How many cannon balls must be fired to bring the car as close to rest as possible?
To answer this question, we need to apply the principle of conservation of momentum. The initial momentum of the system (car, cannon, and cannon balls) is equal to the final momentum.
The momentum of an object is given by the product of its mass and velocity. In this case, the initial momentum of the system is the momentum of the car before any cannon balls are fired, and the final momentum is the momentum of the car after firing a certain number of cannon balls.
Given:
Mass of car (m₁) = 3500 kg
Speed of car (v₁) = 45 m/s
Mass of cannonball (m₂) = 65 kg
Muzzle velocity of cannonball (v₂) = 625 m/s
To find the number of cannon balls required, we need to calculate the final velocity of the car after firing the balls, and then find the number of cannon balls needed to reduce the velocity to zero.
Let's assume the number of cannon balls fired is n.
Using the principle of conservation of momentum:
(m₁ + n * m₂) * v₁ = m₁ * v
Where:
m₁ + n * m₂ = Total mass of the system after firing n cannon balls
v₁ = Initial velocity of the car
v = Final velocity of the car after firing n cannon balls (which we want to be zero)
Rearranging the equation:
v = (m₁ + n * m₂) * v₁ / m₁
To bring the car to rest (v = 0), we can set the equation equal to zero:
(m₁ + n * m₂) * v₁ / m₁ = 0
Simplifying the equation:
(m₁ + n * m₂) = 0
This implies that for the car to come to rest, the total mass of the system after firing n cannon balls should be equal to zero.
Substituting the given values:
3500 + n * 65 = 0
Simplifying the equation:
n * 65 = -3500
Dividing both sides by 65:
n = -3500 / 65
Calculating the value:
n ≈ -53.85
Since we cannot have a negative number of cannon balls, we take the absolute value and round it up:
n = 54
Therefore, to bring the car as close to rest as possible, approximately 54 cannon balls must be fired.