A space vehicle is coasting at a constant velocity of 22.5 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 0.223 m/s2 in the +x direction. After 57.5 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.
I found the answer to part a, but I don't know how to solve part b. Part is 26.1m/s.
To find the magnitude and direction of the vehicle's velocity relative to the space station, we can break down the problem into two components: the initial velocity and the additional velocity due to acceleration.
(a) Magnitude of velocity:
The initial velocity of the vehicle in the +y direction is given as 22.5 m/s. This component remains constant throughout the problem.
The acceleration in the +x direction due to the RCS thruster is given as 0.223 m/s². The time for which the thruster is active is 57.5 s. Using the formula for velocity with constant acceleration:
v = u + a*t
where:
v is the final velocity,
u is the initial velocity,
a is the acceleration,
t is the time elapsed,
we can calculate the additional velocity in the +x direction:
vx = 0 + 0.223 * 57.5
Now, we can use the Pythagorean theorem to find the magnitude of the total velocity:
|v| = sqrt(vx² + vy²)
Given that vy = 22.5 m/s (constant throughout), we can substitute the values:
|v| = sqrt((0.223 * 57.5)² + 22.5²)
Simplifying the equation will give us the magnitude of the velocity.
(b) Direction of velocity:
To find the direction of the velocity relative to the space station, we need to determine the angle (θ) between the positive y-axis and the total velocity vector. We can use trigonometry to find this angle.
tan(θ) = vx / vy
Rearranging the equation gives us:
θ = arctan(vx / vy)
Substituting the respective values of vx and vy will give us the angle in radians. To convert it to degrees, we can multiply by 180/π.
Therefore, by solving these equations, we can find the magnitude and direction of the vehicle's velocity relative to the space station.