A(n) 3800 kg rocket traveling at 2500 m/s is moving freely through space on a journey to the moon. The ground controllers find that the rocket has drifted off course and that it must change direction by 9.2 degrees if it is to hit the moon. By radio control the rocket’s engines are fired instantaneously (i.e., as a single pellet) in a direction perpendicular to that of the rocket’s motion. The gases are expelled (i.e., the pellet) at a speed of 3600 m/s (relative to the rocket). What mass of gas must be expelled to make the needed course correction? Answer in units of kg.
Thank you!
To solve this question, we need to use the principle of conservation of momentum. According to this principle, the total momentum before the engines are fired must be equal to the total momentum after the firing.
Let's break down the different components involved:
1. Initial momentum of the rocket:
The mass of the rocket is given as 3800 kg, and its initial velocity is 2500 m/s. Therefore, the initial momentum of the rocket can be calculated using the formula:
Initial Momentum = Mass × Velocity = 3800 kg × 2500 m/s
2. Final momentum of the rocket:
The rocket needs to change direction by 9.2 degrees, which means that its final velocity will not be in the same direction as the initial velocity. In this case, we can use trigonometry to determine the final velocity in the direction of the moon. The final velocity magnitude can be calculated using the formula:
Final Velocity Magnitude = Initial Velocity × cos(9.2 degrees)
Now, let's calculate the final momentum of the rocket using the formula:
Final Momentum = Mass × Final Velocity Magnitude = 3800 kg × Final Velocity Magnitude
3. Momentum change due to gas expulsion:
The gas is expelled at a speed of 3600 m/s relative to the rocket. This means that the change in momentum due to the expelled gas can be calculated using the formula:
Momentum Change = Mass of Gas × Gas Velocity
Now, since we know that the initial and final momentum of the rocket must be equal, we can set up the equation:
Initial Momentum + Momentum Change = Final Momentum
Plugging in the values, we get:
(3800 kg × 2500 m/s) + (Mass of Gas × 3600 m/s) = (3800 kg × Final Velocity Magnitude)
Solving the equation for the Mass of Gas will give us the answer.
Note: Since the question asks for the mass of gas in kg, we will leave the velocity values in m/s to maintain consistent units of measurement.
To solve this problem, we can use the law of conservation of linear momentum.
The initial momentum of the rocket before firing the engines is given by:
P_initial = m_initial * v_initial
where m_initial is the initial mass of the rocket and v_initial is its initial velocity.
The final momentum of the rocket after firing the engines is given by:
P_final = m_final * v_final
where m_final is the final mass of the rocket (including the expelled gas) and v_final is its final velocity.
According to the law of conservation of linear momentum, the initial momentum of the rocket must be equal to the final momentum of the rocket:
P_initial = P_final
Let's first calculate the initial momentum of the rocket:
P_initial = m_initial * v_initial
= (3800 kg) * (2500 m/s)
Now, let's calculate the final momentum of the rocket:
P_final = m_final * v_final
= (m_initial + m_gas) * (v_initial + v_gas)
= (3800 kg + m_gas) * (2500 m/s + 3600 m/s)
= (3800 kg + m_gas) * (6100 m/s)
Since the rocket changes direction by 9.2 degrees, the angle between the initial and final velocities is 90 degrees.
Using the law of conservation of linear momentum, we can write:
P_initial = P_final
m_initial * v_initial = (3800 kg + m_gas) * (6100 m/s)
Now, let's solve for m_gas:
m_gas = (m_initial * v_initial) / (6100 m/s) - 3800 kg
Substituting the given values:
m_gas = (3800 kg * 2500 m/s) / (6100 m/s) - 3800 kg
Simplifying the equation:
m_gas = 977.05 kg
Therefore, the mass of gas that must be expelled to make the needed course correction is approximately 977.05 kg.