The mass of a rocket car plus fuel is 2,000 kg. The rocket car starts from rest. The engine expels fuel (in the form of exhaust) over a period of 8 seconds. The exhaust has a velocity of 1,000 m/s due west, and the velocity of the rocket after 8 seconds is 90 m/s due east. How much fuel did the rocket car exhaust during the 8 seconds? (Answer is in kg)
momentum is conserved
1000 m/s * x = (2000 - x) * 90 m/s
Thank you sm!!! I have been struggling this for a while and I overcomplicated it!
To solve this problem, we need to apply the principle of conservation of momentum.
The initial momentum of the system (rocket car plus fuel) is given by:
Initial momentum = mass * initial velocity
The final momentum of the system is given by:
Final momentum = mass * final velocity
According to the principle of conservation of momentum, the initial momentum should be equal to the final momentum.
Since the rocket car starts from rest, the initial velocity is 0 m/s. Therefore:
Initial momentum = 0
The final velocity of the rocket car is given as 90 m/s due east. Therefore:
Final momentum = (2000 kg + fuel mass) * 90 m/s
Since the rocket car engine expels fuel with a velocity of 1000 m/s due west, the momentum of the expelled fuel is given by:
Momentum of expelled fuel = fuel mass * (−1000 m/s)
Since the total momentum of the system is conserved:
Initial momentum + Momentum of expelled fuel = Final momentum
Therefore,
0 + (fuel mass * −1000 m/s) = (2000 kg + fuel mass) * 90 m/s
Let's solve this equation step by step:
0 - 1000 * fuel mass = 90 * (2000 + fuel mass)
-1000 * fuel mass = 90 * 2000 + 90 * fuel mass
-1000 * fuel mass = 180000 + 90 * fuel mass
-1000 * fuel mass - 90 * fuel mass = 180000
-1090 * fuel mass = 180000
fuel mass = 180000 / -1090
fuel mass ≈ 165.14 kg
Therefore, the rocket car exhausted approximately 165.14 kg of fuel during the 8 seconds.
To determine the amount of fuel exhausted by the rocket car during the 8 seconds, we can use the principle of conservation of momentum.
The momentum of an object is defined as the product of its mass and velocity. The law of conservation of momentum states that the total momentum of a system of objects remains constant if no external forces act on it.
Initially, the rocket car is at rest, so its initial momentum is zero. After 8 seconds, the rocket car has a final velocity of 90 m/s due east. The exhaust, on the other hand, has a velocity of 1000 m/s due west.
Let's assume the mass of the exhausted fuel is m kg.
According to the conservation of momentum, the total momentum before the exhaust is equal to the total momentum after the exhaust.
Initial momentum = Final momentum
(0 kg) * (0 m/s) + (m kg) * (1000 m/s) = (2000 kg) * (90 m/s)
Simplifying the equation:
1000m = 180000 kg * m/s
Dividing both sides of the equation by 1000:
m = 180 kg
Therefore, the rocket car exhausted 180 kg of fuel during the 8 seconds.