A 4.40-kg model rocket is launched, shooting 53.0 g of burned fuel from its exhaust at an average velocity of 650 m/s. What is the velocity of the rocket after the fuel has burned? (Ignore effects of gravity and air resistance.)
To find the velocity of the rocket after the fuel has burned, we can use the concept of conservation of momentum. The initial momentum of the rocket and fuel system is equal to the final momentum after the fuel is burned.
The initial momentum (before burning the fuel) is given by the product of the total mass and the initial velocity:
Initial momentum = (Mass of rocket + Mass of fuel) × Initial velocity
The final momentum (after the fuel is burned) is given by the product of the mass of the rocket and the final velocity:
Final momentum = Mass of rocket × Final velocity
According to the law of conservation of momentum, these two momenta should be equal:
Initial momentum = Final momentum
Let's calculate the values:
Mass of rocket = 4.40 kg
Mass of fuel = 53.0 g = 0.053 kg
Initial velocity = ?
Final velocity = ?
Now, let's substitute these values into the equation of conservation of momentum:
(Mass of rocket + Mass of fuel) × Initial velocity = Mass of rocket × Final velocity
(4.40 kg + 0.053 kg) × Initial velocity = 4.40 kg × Final velocity
Simplifying the equation:
4.453 kg × Initial velocity = 4.40 kg × Final velocity
We can solve this equation for the Final velocity:
Final velocity = (4.453 kg × Initial velocity) / 4.40 kg
Given that the fuel is burned at an average velocity of 650 m/s, we can use this value as the Initial velocity:
Final velocity = (4.453 kg × 650 m/s) / 4.40 kg
The units of kg cancel out, and we're left with the final velocity in m/s:
Final velocity = (4.453 × 650) / 4.40 m/s
Simplifying the calculation:
Final velocity ≈ 672.95 m/s
Therefore, the final velocity of the rocket after the fuel has burned is approximately 672.95 m/s.