A model rocket is launched at 62 degrees above the horizontal with an initial speed of 28m/s. the rocket travels for 4.5s along its initial line of motion with a constant acceleration of 13 m/s^2. At this time, the engine of the rocket stops and the rocket proceeds to move as a projectile. Ignore asked resistance.
Find the maximum altitude the rocket reaches. I got 31.2 using the v=vo^2 +2ad formula with v=0, v0= 28sin62, and a=-9.8. However the answer key says 530m. What did I do wrong?
Also, how would I find the total time the rocket is in the air?
To find the maximum altitude the rocket reaches, you need to consider two parts of its motion: when it is under constant acceleration and when it becomes a projectile after the engine stops.
1. Constant acceleration phase:
You correctly identified that the rocket travels for 4.5 seconds with a constant acceleration of 13 m/s^2. To find the vertical displacement during this phase, you can use the formula:
y = y0 + v0y * t + 0.5 * a * t^2,
where y is the vertical displacement, y0 is the initial vertical position, v0y is the vertical component of the initial velocity, t is the time, and a is the acceleration.
Here, y0 = 0 (assuming the ground is the reference level), v0y = (28 m/s) * sin(62°) ≈ 24.09 m/s, a = -9.8 m/s^2 (negative due to gravity), and t = 4.5 s.
Plugging in these values, we have:
y = 0 + (24.09 m/s) * (4.5 s) + 0.5 * (-9.8 m/s^2) * (4.5 s)^2
y ≈ 54.61 m.
2. Projectile motion phase:
After the engine stops, the rocket becomes a projectile moving under the influence of gravity. The time it remains in the air can be found using the vertical component of its initial velocity and the acceleration due to gravity. We can use the formula:
v = v0 + a * t,
where v is the final vertical velocity, v0 is the initial vertical velocity, a is the acceleration, and t is the time.
Here, v0 = v0y ≈ 24.09 m/s, v = 0 (at the maximum height), and a = -9.8 m/s^2.
Plugging in these values, we have:
0 = 24.09 m/s + (-9.8 m/s^2) * t
t ≈ 2.46 s.
Now, to find the total time the rocket is in the air, we can add the times from the constant acceleration phase and the projectile motion phase:
Total time = 4.5 s + 2.46 s ≈ 6.96 s.
Regarding the discrepancy in the maximum altitude, it seems you used the wrong value for the acceleration due to gravity. The correct value is -9.8 m/s^2, not 13 m/s^2. Using the correct value should yield a more accurate result.