A rocket takes off from Earth and reaches a speed of 100 m/s in 10.0 s. If the exhaust speed is 1500 m/s and the mass of fuel burned is 100 kg, what was the initial mass of the rocket?
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To find the initial mass of the rocket, we will use the principle of conservation of momentum. The momentum before the rocket takes off is equal to the momentum after it reaches the speed of 100 m/s.
The momentum before takeoff is given by the product of the initial mass of the rocket (m) and the initial velocity (v), which is initially at rest, so the initial velocity is 0 m/s.
The momentum after reaching a speed of 100 m/s is given by the product of the final mass of the rocket (m') and the final velocity (v').
According to the principle of conservation of momentum, the momentum before takeoff is equal to the momentum after reaching 100 m/s:
mv = m'v'
Since the rocket undergoes propulsion by expelling a mass of fuel, we can determine the change in momentum using the rocket equation:
Δp = -v * Δm
Where Δp is the change in momentum, v is the exhaust velocity, and Δm is the mass of fuel burned.
In this case, we have:
Δp = -1500 m/s * 100 kg
Simplifying:
Δp = -150,000 kg*m/s
We also know that:
m' = m - Δm
Substituting this into the conservation of momentum equation:
mv = (m - Δm) * 100 m/s
Expanding and rearranging the equation:
100 m^2/s = 100 m - Δm * 100 m/s
Dividing both sides by 100 m/s:
m = 100 m - Δm
Substituting the value of Δm from the rocket equation:
m = 100 m + 150,000 kg
Simplifying:
m = 150,100 kg
Therefore, the initial mass of the rocket was 150,100 kg.
To solve this problem, we can use the principle of conservation of momentum. The momentum before the rocket takes off is equal to the momentum after it reaches a speed of 100 m/s.
Step 1: Calculate the momentum before the rocket takes off.
The momentum before the rocket takes off is given by the equation: momentum = mass × velocity.
Let's assume the initial mass of the rocket is M (kg).
The momentum before takeoff is 0 because the rocket is stationary at that time.
Momentum before takeoff = mass × velocity
= M × 0
= 0
Step 2: Calculate the momentum after the rocket reaches a speed of 100 m/s.
The momentum after the rocket reaches a speed of 100 m/s is given by the equation: momentum = mass × velocity.
The mass of the rocket after burning 100 kg of fuel is (M - 100) kg.
The final velocity is 100 m/s.
Momentum after reaching 100 m/s = (M - 100) × 100
Step 3: Apply the principle of conservation of momentum.
According to the principle of conservation of momentum, the momentum before and after the rocket takes off should be equal.
0 = (M - 100) × 100
Simplify the equation:
0 = 100M - 10000
Step 4: Solve for M.
Rearrange the equation:
100M = 10000
Divide both sides by 100:
M = 10000 / 100
= 100 kg
Therefore, the initial mass of the rocket was 100 kg.