Two rockets are flying in the same direction and are side by side at the instant their retrorockets fire. Rocket A has an initial velocity of +5800 m/s, while rocket B has an initial velocity of +9100 m/s. After a time t both rockets are again side by side, the displacement of each being zero. The acceleration of rocket A is -12 m/s2. What is the acceleration of rocket B?
To find the acceleration of rocket B, we can use the equation of motion relating displacement, initial velocity, acceleration, and time:
\[s = ut + \frac{1}{2}at^2\]
In this case, since both rockets start side by side and end up side by side with zero displacement, their displacements are both zero.
For Rocket A:
Initial velocity (u) = +5800 m/s
Acceleration (a) = -12 m/s^2
Displacement (s) = 0
Applying these values to the equation of motion, we get:
\[0 = (5800) t + \frac{1}{2}(-12) t^2\]
Now let's solve this quadratic equation for the value of time (t).
First, let's multiply everything by 2 to get rid of the fraction:
\[0 = (11600) t + (-12) t^2\]
Rearranging the equation, we get:
\[12t^2 - 11600t = 0\]
We can factor the equation:
\[t(12t - 11600) = 0\]
So, either t = 0 or (12t - 11600) = 0
Since we are looking for a positive time (t), t = 0 is not a valid solution.
Therefore, we have: (12t - 11600) = 0
Solving for t, we get:
12t = 11600
t = 11600 / 12
t ≈ 966.667 s
Now that we have the value of time (t), we can find the acceleration of Rocket B using the equation of motion:
s = ut + 1/2at^2
For Rocket B, initial velocity (u) = +9100 m/s, displacement (s) = 0, and we've already determined that t = 966.667 s.
Substituting the values into the equation:
0 = (9100) (966.667) + 1/2(a) (966.667)^2
Simplifying the equation:
0 = 8796333.3 + 0.5a(933607.782)
Rearranging the equation:
-8796333.3 = 0.5a(933607.782)
Dividing both sides by 0.5(933607.782):
a ≈ -18.809 m/s^2
Therefore, the acceleration of rocket B is approximately -18.809 m/s^2.