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.

Question

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!

Answer 1

To answer this question, we can use the principle of conservation of momentum. The momentum of the rocket (before the course correction) must be equal to the momentum of the rocket plus the expelled gas (after the course correction).

First, let's find the momentum of the rocket before the course correction. Momentum is defined as the product of mass and velocity. So, the momentum of the rocket before the course correction is given by:

Momentum_rocket_before = mass_rocket * velocity_rocket_before

Substituting the given values:
Momentum_rocket_before = 3800 kg * 2500 m/s

Next, let's find the momentum of the rocket plus the expelled gas after the course correction. Since the rocket fires the gas pellet in a direction perpendicular to its motion, the initial momentum of the expelled gas is zero. The final momentum after the course correction is given by:

Momentum_rocket_plus_gas = (mass_rocket + mass_gas) * velocity_rocket_plus_gas

The initial angle between the rocket's motion and the direction of the expelled gas velocity is 9.2 degrees. To find the velocity of the rocket plus the expelled gas, we can use simple trigonometry:

velocity_rocket_plus_gas = √(velocity_rocket_before^2 + velocity_expelled_gas^2 - 2 * velocity_rocket_before * velocity_expelled_gas * cos(initial_angle))

Substituting the given values:
velocity_rocket_plus_gas = √((2500 m/s)^2 + (3600 m/s)^2 - 2 * 2500 m/s * 3600 m/s * cos(9.2 degrees))

Now, we can equate the momentum before and after the course correction:

mass_rocket * velocity_rocket_before = (mass_rocket + mass_gas) * velocity_rocket_plus_gas

Since we want to find the mass of the gas expelled, we can rearrange the equation to solve for mass_gas:

mass_gas = (mass_rocket * velocity_rocket_before - (mass_rocket + mass_gas) * velocity_rocket_plus_gas) / velocity_rocket_plus_gas

Now, we can substitute the values and calculate the mass of the gas:

mass_gas = (3800 kg * 2500 m/s - (3800 kg + mass_gas) * velocity_rocket_plus_gas) / velocity_rocket_plus_gas

We can solve this equation iteratively to find the value of mass_gas.