A 19 N thrust is exerted on a 0.4 kg rocket for a time interval of 8 s as it moves straight up. (Neglect all forces except this thrust and the force of gravity.) The rocket then continues to move upward as its speed reduces steadily to zero.
(a) What is the maximum height that the rocket attains?
To find the maximum height that the rocket attains, we can break down the problem into two parts: the initial upward motion with constant acceleration and the subsequent upward motion with decreasing speed.
First, let's calculate the acceleration of the rocket during the initial phase. We can use Newton's second law of motion:
Force = mass x acceleration
The only force acting on the rocket during the initial phase is the thrust force, which is given as 19 N. The mass of the rocket is given as 0.4 kg. Therefore, we can rearrange the equation to solve for acceleration:
Acceleration = Force / Mass
Acceleration = 19 N / 0.4 kg
Acceleration = 47.5 m/s^2
Now that we know the acceleration, we can determine the upward velocity of the rocket at the end of the initial phase. We'll use the kinematic equation:
Final velocity = Initial velocity + (acceleration x time)
Since the rocket starts at rest, the initial velocity is 0. The time interval for the initial phase is given as 8 s. Plugging these values into the equation:
Final velocity = 0 + (47.5 m/s^2 x 8 s)
Final velocity = 380 m/s
Now, we can use this final velocity to calculate the maximum height reached by the rocket during the second phase. In this phase, the rocket is slowing down until its velocity reaches 0 m/s.
To find the maximum height, we can use another kinematic equation:
Final velocity^2 = Initial velocity^2 + (2 x acceleration x distance)
Since the rocket's final velocity is 0 m/s and the initial velocity is 380 m/s, we can solve for distance:
0^2 = 380^2 + (2 x (-9.8 m/s^2) x distance)
0 = 144400 - 19.6d
19.6d = 144400
d = 144400 / 19.6
d ≈ 7367.3 m
Therefore, the maximum height that the rocket attains is approximately 7367.3 meters.