You have designed a rocket to be used to sample the local atmosphere for pollution. It is fired vertically with a constant upward acceleration of 17 m/s2. After 20 s, the engine shuts off and the rocket continues rising (in freefall) for a while. (Neglect any effects due to air resistance.) The rocket eventually stops rising and then falls back to the ground. You want to get a sample of air that is 12 km above the ground.
Vo = a*t = 17m/s^2 * 20s = 340 m/s.
h1 = 0.5a*t^2 = 0.5*17*20^2 = 3400 m. = Ht. at which the engine shuts off.
hmax = h1 + (V^2-Vo^2)/2g.
hnax = 3400 + (0-(340^2)/-19.6 =9298 m. = 9.298 km above gnd.
The rocket did not reach the required 12 km.
To find the time it takes for the rocket to reach a height of 12 km above the ground, we can break down the problem into two parts: the time it takes for the rocket to reach its maximum height, and the time it takes for the rocket to fall back to the ground.
First, let's find the time it takes for the rocket to reach its maximum height. We can use the kinematic equation:
vf = vi + at
Where:
- vf is the final velocity
- vi is the initial velocity
- a is the acceleration
- t is the time
Since the rocket is fired vertically with a constant upward acceleration of 17 m/s^2 and starts from rest (vi = 0), we can rearrange the equation to solve for the time it takes to reach the maximum height:
t = vf / a
We know that the final velocity at the maximum height is 0, so we can substitute it into the equation:
t = 0 / 17
t = 0 seconds
This means that it takes 0 seconds for the rocket to reach its maximum height. However, this situation doesn't make physical sense, so we need to reevaluate our approach.
Since the rocket is in freefall after the engine shuts off, it will eventually reach a point where its upward velocity is reduced to 0. At this point, gravity will start pulling the rocket downward, causing it to fall back to the ground.
To find the time it takes for the rocket to reach a height of 12 km above the ground, we can set up the following equation:
12 km = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the rocket experiences freefall, the initial velocity is 0, and the acceleration due to gravity is approximately -9.8 m/s^2 (taking into account the negative direction). Converting the height to meters, we have:
12 km = (0 * time) + (0.5 * -9.8 * time^2)
12000 m = -4.9 * time^2
Rearranging the equation, we get:
time^2 = 12000 / -4.9
time^2 = -2448.98
Taking the square root of both sides, we find:
time = √(-2448.98)
The problem presents a logical inconsistency: the time needed for the rocket to reach a height of 12 km above the ground cannot be determined with the given information. It is possible that a key value is missing or the problem is not adequately described.