On a spacecraft, two engines are turned on for 910 s at a moment when the velocity of the craft has x and y components of v0x = 5710 m/s and v0y = 7490 m/s. While the engines are firing, the craft undergoes a displacement that has components of x = 5.78 x 106 m and y = 6.17 x 106 m. Find the (a) x and (b) y components of the craft's acceleration.
To find the x and y components of the craft's acceleration, we can use the following kinematic equation:
Δv = v0 + at
Where:
Δv is the change in velocity
v0 is the initial velocity
a is the acceleration
t is the time
For the x-component of acceleration, we will use the displacement, initial velocity, and time in the x-direction. Similarly, for the y-component of acceleration, we will use the displacement, initial velocity, and time in the y-direction.
(a) x-component of acceleration:
Given:
Initial velocity in the x-direction (v0x) = 5710 m/s
Displacement in the x-direction (x) = 5.78 x 10^6 m
Using the equation Δv = v0 + at, we rearrange it to solve for acceleration (a):
a = (Δv - v0) / t
Since the spacecraft is initially at rest in the x-direction, Δv in the x-direction is equal to the final velocity (vfx) in the x-direction:
vfx = v0x + ax * t
Since we are given the time (t) for which the engines are turned on (910 s), we can substitute the values to calculate the x-component of the craft's acceleration.
(b) y-component of acceleration:
Given:
Initial velocity in the y-direction (v0y) = 7490 m/s
Displacement in the y-direction (y) = 6.17 x 10^6 m
Using the same equation, Δv = v0 + at, and solving for acceleration (a):
a = (Δv - v0) / t
Again, since the spacecraft is initially at rest in the y-direction, Δv in the y-direction is equal to the final velocity (vfy) in the y-direction:
vfy = v0y + ay * t
Substituting the given values and the time (t), we can calculate the y-component of the craft's acceleration.
By solving these two equations, we can find the x and y components of the craft's acceleration.