A spacecraft is traveling with a velocity of v0x = 6370 m/s along the +x direction. Two engines are turned on for a time of 843 s. One engine gives the spacecraft an acceleration in the +x direction of ax = 1.20 m/s2, while the other gives it an acceleration in the +y direction of ay = 7.80 m/s2. At the end of the firing, find the following.
a). vy and vx
Acceleration (along any axis) times the time interval that it acts equals velocity change (along that axis)
Vx increases by an amount 1.2 m/s^2*843 s. That adds 1012 m/s to the vx component.
The Vy component starts from zero and ends up at 7.8 m/s^2*843 s = 6575 m/s
To find the final velocities vy and vx of the spacecraft, we first need to calculate the change in velocity caused by each engine separately and then combine them.
Let's start with the engine that gives the spacecraft an acceleration in the +x direction.
We know the initial velocity v0x = 6370 m/s, and the acceleration ax = 1.20 m/s². The time of firing is given as 843 s.
Using the formula: vf = v0x + ax * t, where vf is the final velocity, v0x is the initial velocity, ax is the acceleration, and t is the time, we can substitute the values:
vf_x = v0x + ax * t
vf_x = 6370 m/s + (1.20 m/s²) * 843 s
vf_x = 6370 m/s + 1011.6 m/s
vf_x = 7381.6 m/s
So, the final velocity in the +x direction is vf_x = 7381.6 m/s.
Now let's find the final velocity in the +y direction caused by the second engine.
We know the initial velocity in the y-direction is zero (v0y = 0 m/s), and the acceleration in the y-direction is ay = 7.80 m/s². The time of firing is again given as 843 s.
Using the same formula as before: vf = v0y + ay * t, and substituting the values:
vf_y = v0y + ay * t
vf_y = 0 m/s + (7.80 m/s²) * 843 s
vf_y = 0 m/s + 6572.4 m/s
vf_y = 6572.4 m/s
So, the final velocity in the +y direction is vf_y = 6572.4 m/s.
Therefore, the final velocities of the spacecraft are:
vx = 7381.6 m/s in the +x direction
vy = 6572.4 m/s in the +y direction.