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.
Vox = 5710 m/s
Voy = 7490 m/s
a. Hor. acceleration:
Vo*t + 0.5a*t^2 = 5.78*10^6 m.
5710*910 + 0.5a*910^2 = 5.78*10^6
5.196*10^6 + 4.14*10^5a = 5.78*10^6
4.14*10^5a = 5.84*10^5
a = 1.41 m/s^2.
b. Ver. acceleration:
7490*910 + 0.5a*910^2 = 6.17*10^6
6.816*10^6 + 4.14*10^5a = 6.17*10^6
4.14*10^5a = -6.46*10^5
a = -1.56 m/s^2.
To find the x and y components of the spacecraft's acceleration, we can use the kinematic equation:
Displacement = Initial velocity * Time + (1/2) * Acceleration * Time^2.
Given:
Initial velocity: v0x = 5710 m/s (x component) and v0y = 7490 m/s (y component).
Displacement: x = 5.78 x 10^6 m (x component) and y = 6.17 x 10^6 m (y component).
Time: t = 910 s.
We can use these values to find the accelerations in the x and y directions separately.
(a) Finding the x-component of acceleration:
Using the given values, the equation becomes:
5.78 x 10^6 m = (5710 m/s) * (910 s) + (1/2) * ax * (910 s)^2.
Simplifying the equation:
5.78 x 10^6 m = 5197100 m + (1/2) * ax * 828100 s^2.
Rearranging the equation:
(1/2) * ax * 828100 s^2 = 5.78 x 10^6 m - 5197100 m.
(1/2) * ax * 828100 s^2 = 1092900 m.
ax * 828100 s^2 = 1092900 m.
ax = 1092900 m / (828100 s^2).
Calculating ax:
ax = 1.3204 m/s^2.
Therefore, the x component of the spacecraft's acceleration is 1.3204 m/s^2.
(b) Finding the y-component of acceleration:
Using the given values, the equation becomes:
6.17 x 10^6 m = (7490 m/s) * (910 s) + (1/2) * ay * (910 s)^2.
Simplifying the equation:
6.17 x 10^6 m = 6809900 m + (1/2) * ay * 828100 s^2.
Rearranging the equation:
(1/2) * ay * 828100 s^2 = 6.17 x 10^6 m - 6809900 m.
(1/2) * ay * 828100 s^2 = 48010 m.
ay * 828100 s^2 = 48010 m.
ay = 48010 m / (828100 s^2).
Calculating ay:
ay = 0.058 m/s^2.
Therefore, the y component of the spacecraft's acceleration is 0.058 m/s^2.
So, the x and y components of the spacecraft's acceleration are:
(a) x acceleration = 1.3204 m/s^2.
(b) y acceleration = 0.058 m/s^2.