A rocket is launched with a speed of 100m/s at an angle of 53.0 degrees above the horizontal with an acceleration of 30 metres per square seconds. After 3 seconds it's engines fail and the rocket proceeds to move as a projectile.Find the maximum height reached,the horizontal distance covered and the total time taken to cover this distance.
To solve this problem, we can break it down into three parts:
1. The rocket's motion while the engines are working.
2. The rocket's motion after the engines fail, while it moves as a projectile.
3. Finding the maximum height reached, horizontal distance covered, and total time taken.
1. The rocket's motion while the engines are working:
The initial speed of the rocket is 100 m/s, and it is launched at an angle of 53.0 degrees above the horizontal. Let's break down the initial velocity into horizontal and vertical components.
- The horizontal component of the initial velocity (Vx) can be calculated using Vx = V * cos(θ), where V is the initial speed and θ is the launch angle.
Vx = 100 m/s * cos(53) = 100 m/s * 0.6 = 60 m/s.
- The vertical component of the initial velocity (Vy) can be calculated using Vy = V * sin(θ).
Vy = 100 m/s * sin(53) = 100 m/s * 0.8 = 80 m/s.
2. The rocket's motion after the engines fail:
After 3 seconds, the rocket's engines fail, and it continues to move as a projectile. We know that acceleration due to gravity is -9.8 m/s^2 (negative because it acts downwards). The rocket will continue to move along the horizontal direction at a constant speed, while the vertical motion will be influenced by gravity.
- The time taken for the projectile motion (t) is the total time minus the initial time (t_initial).
t = 3 seconds (total time) - 0 seconds (initial time) = 3 seconds.
- The displacement in the horizontal direction (Sx) can be calculated using Sx = Vx * t.
Sx = 60 m/s * 3 s = 180 m.
- In the vertical direction, the motion will be influenced by gravity. The maximum height (H) reached by the rocket can be calculated using H = Vy * t + (1/2) * a * t^2, where a is the acceleration due to gravity.
H = 80 m/s * 3 s + (1/2) * (-9.8 m/s^2) * (3 s)^2 = 240 m - 44.1 m = 195.9 m.
3. Finding the maximum height reached, horizontal distance covered, and total time taken:
From the previous calculations, we have:
- Maximum height reached (H) = 195.9 m.
- Horizontal distance covered (Sx) = 180 m.
- Total time taken (t) = 3 seconds.
Therefore, the maximum height reached by the rocket is 195.9 meters, the horizontal distance covered is 180 meters, and the total time taken to cover this distance is 3 seconds.