A space vehicle is coasting at a constant velocity of 24.9 m/s in the +y direction relative to a space station. The pilot of the vehicle fires a RCS (reaction control system) thruster, which causes it to accelerate at 0.338 m/s2 in the +x direction. After 53.7 s, the pilot shuts off the RCS thruster. After the RCS thruster is turned off, find (a) the magnitude and (b) the direction of the vehicle's velocity relative to the space station. Express the direction as an angle (in degrees) measured from the +y direction.
To find the magnitude and direction of the vehicle's velocity relative to the space station after the RCS thruster is turned off, we can use the principles of relative motion.
Given information:
Initial velocity of the vehicle, v0 = 24.9 m/s in the +y direction
Acceleration due to RCS thruster, a = 0.338 m/s² in the +x direction
Time when RCS thruster is turned off, t = 53.7 s
Step 1: Calculate the change in velocity (Δv) caused by the RCS thruster.
Δv = a * t
Δv = 0.338 m/s² * 53.7 s
Δv ≈ 18.1396 m/s in the +x direction
Step 2: Calculate the final velocity of the vehicle relative to the space station by adding the change in velocity to the initial velocity vector.
Final velocity, v = v0 + Δv
For the magnitude |v|:
|v| = √(vx² + vy²)
The x-component of the final velocity (vx) is the sum of the initial x-component velocity and the change in velocity:
vx = 0 m/s + 18.1396 m/s = 18.1396 m/s
The y-component of the final velocity (vy) remains the same:
vy = 24.9 m/s
|v| = √(18.1396² + 24.9²)
|v| ≈ 30.053 m/s
The magnitude of the vehicle's final velocity relative to the space station is approximately 30.053 m/s.
Step 3: Calculate the direction of the vehicle's velocity relative to the space station.
Use the inverse tangent function to find the angle (θ) measured from the +y direction:
θ = tan^(-1)(vx / vy)
θ = tan^(-1)(18.1396 m/s / 24.9 m/s)
θ ≈ 37.175°
The direction of the vehicle's velocity relative to the space station is approximately 37.175° measured from the +y direction.