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 principle of conservation of momentum. According to this principle, the total momentum before the fuel is burned is equal to the total momentum after the fuel is burned.
The momentum of an object can be calculated by multiplying its mass by its velocity. So, we can calculate the momentum of the rocket before and after the fuel is burned.
Let's denote the velocity of the rocket after the fuel is burned as Vf and the velocity of the burned fuel as Vb.
Given:
Mass of the rocket (m1) = 4.40 kg
Mass of the burned fuel (m2) = 53.0 g = 0.053 kg
Velocity of the burned fuel (Vb) = 650 m/s
Before the fuel is burned, the rocket and the fuel are both traveling with the same velocity, let's say Vo.
The total momentum before the fuel is burned is:
P1 = (m1 + m2) * Vo
After the fuel is burned, the rocket is traveling with velocity Vf. The momentum of the rocket is:
P2 = m1 * Vf
The fuel is shot backward with a velocity Vb, so its momentum is:
P3 = m2 * (-Vb) (Negative sign because the velocity is in the opposite direction)
According to the principle of conservation of momentum:
P1 = P2 + P3
Substituting the values, we get:
(m1 + m2) * Vo = m1 * Vf - m2 * Vb
Simplifying the equation by rearranging the terms:
(m1 + m2) * Vo = m1 * Vf + (-m2) * Vb
Using the fact that the velocity of the fuel is in the opposite direction, we can write Vb as -Vb:
(m1 + m2) * Vo = m1 * Vf - m2 * Vb
Now, we can solve this equation for Vf:
(m1 + m2) * Vo + m2 * Vb = m1 * Vf
Plugging in the values:
(4.40 kg + 0.053 kg) * Vo + 0.053 kg * (-650 m/s) = 4.40 kg * Vf
Simplifying further and solving for Vf:
4.453 kg * Vo - 0.03445 kg*m/s = 4.40 kg * Vf
Vf = (4.453 kg * Vo - 0.03445 kg*m/s) / 4.40 kg
Therefore, the velocity of the rocket after the fuel has burned is (4.453 kg * Vo - 0.03445 kg*m/s) / 4.40 kg.