A model rocket is launched straight upward with an initial speed of 60.0 m/s. It accelerates with a constant upward acceleration of 2.00 m/s2 until its engines stop at an altitude of 130 m.

(a) What is the maximum height reached by the rocket?


(b) How long after lift-off does the rocket reach its maximum height?
s

(c) How long is the rocket in the air?
s

To find the answers to these questions, we can use the kinematic equations of motion. Let's start with the first question:

(a) What is the maximum height reached by the rocket?

To find the maximum height, we need to find the time it takes for the rocket to reach that point. We can use the equation:

vf^2 = vi^2 + 2as

where vf is the final velocity, vi is the initial velocity, a is the acceleration, and s is the displacement.

In this case, the initial velocity (vi) is 60.0 m/s, the acceleration (a) is 2.00 m/s^2, and the displacement (s) is the maximum height reached by the rocket. We can assume that the final velocity (vf) is 0 since the rocket stops when it reaches the maximum height.

0 = (60.0 m/s)^2 + 2(2.00 m/s^2)s

Simplifying the equation:

0 = 3600 m^2/s^2 + 4.00 m/s^2s

Now, we can rearrange the equation to solve for s:

-3600 m^2/s^2 = 4.00 m/s^2s

Simplifying further:

-3600 m^2/s^2 / 4.00 m/s^2 = s

The meters squared per second squared (m^2/s^2) units cancel out, and we get:

-900 s = s

Therefore, the rocket reaches a maximum height of 900 meters.

(b) How long after lift-off does the rocket reach its maximum height?

To find the time it takes for the rocket to reach its maximum height, we can use another kinematic equation:

vf = vi + at

where t is the time.

In this case, the initial velocity (vi) is 60.0 m/s, the acceleration (a) is 2.00 m/s^2, and the final velocity (vf) is 0 (since the rocket stops). We need to solve for t:

0 = 60.0 m/s + 2.00 m/s^2t

Rearranging the equation:

-60.0 m/s = 2.00 m/s^2t

Dividing both sides by 2.00 m/s^2:

-30.0 s = t

Therefore, the rocket reaches its maximum height 30.0 seconds after lift-off.

(c) How long is the rocket in the air?

To find the total time the rocket is in the air, we need to determine when it lands. Since the rocket reaches its maximum height and then comes back down to the ground, we can double the time it takes to reach the maximum height (from part b).

Total time = 2 * 30.0 s = 60.0 s

Therefore, the rocket is in the air for 60.0 seconds.