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.
See previous post.
To find the x and y components of the spacecraft's acceleration, we need to use kinematic equations.
The kinematic equation that relates acceleration, initial velocity, displacement, and time is:
d = v0t + (1/2)at^2
where:
d = displacement
v0 = initial velocity
t = time
a = acceleration
We can use this equation separately for the x and y components.
(a) Finding the x-component acceleration:
Given:
Initial velocity in the x direction (v0x) = 5710 m/s
Displacement in the x direction (x) = 5.78 x 10^6 m
Time (t) = 910 s
We can rearrange the equation to solve for acceleration (a):
x = v0xt + (1/2)at^2
Rearrange the equation:
a = (2x - 2v0xt) / t^2
Substitute in the given values:
a_x = (2 * 5.78 x 10^6 m - 2 * 5710 m/s * 910 s) / (910 s)^2
Calculating the value:
a_x = (2 * 5.78 x 10^6 m - 2 * 5710 m/s * 910 s) / 828100 s^2
Simplifying the equation:
a_x = 1.33 m/s^2
Therefore, the x-component of the spacecraft's acceleration is 1.33 m/s^2.
(b) Finding the y-component acceleration:
Given:
Initial velocity in the y direction (v0y) = 7490 m/s
Displacement in the y direction (y) = 6.17 x 10^6 m
Time (t) = 910 s
Using the same formula rearranged for the y direction:
y = v0yt + (1/2)at^2
Rearrange the equation:
a = (2y - 2v0yt) / t^2
Substitute in the given values:
a_y = (2 * 6.17 x 10^6 m - 2 * 7490 m/s * 910 s) / (910 s)^2
Calculating the value:
a_y = (2 * 6.17 x 10^6 m - 2 * 7490 m/s * 910 s) / 828100 s^2
Simplifying the equation:
a_y = 3.57 m/s^2
Therefore, the y-component of the spacecraft's acceleration is 3.57 m/s^2.
Summary:
(a) The x-component of the spacecraft's acceleration is 1.33 m/s^2.
(b) The y-component of the spacecraft's acceleration is 3.57 m/s^2.