A rocket is launched at an angle of è = 48° above the horizontal with an initial speed vi = 62 m/s, as shown below. It moves for 25 s along its initial line of motion wth an acceleration of 25.6 m/s2. At this time, its engines fail and the rocket proceeds to move as a free body.
(a) What is the rocket's maximum altitude?
(b) What is the rocket's total time of flight?
(c) What is the rocket's horizontal range?
To solve this problem, we can break it down into different stages and use the equations of motion and projectile motion to find the answers.
1. Find the initial vertical velocity:
The given initial speed vi can be divided into its vertical and horizontal components. The vertical component is given by vi * sin(è). Therefore, the initial vertical velocity vyi = vi * sin(48°).
2. Find the time it takes for the rocket's engines to fail:
Given that the rocket moves for 25 seconds along its initial line of motion, the time it takes for the engines to fail is 25 seconds.
3. Find the maximum altitude:
When the rocket's engines fail, it becomes a projectile motion problem. The vertical component of motion can be described using the equation:
y = y0 + vyi * t - (1/2) * g * t^2
where y0 is the initial vertical position (which is 0 as the rocket starts from the ground), vyi is the initial vertical velocity, t is the time, and g is the acceleration due to gravity.
At the maximum altitude, the vertical velocity becomes zero. Therefore, we can set vyi * t - (1/2) * g * t^2 = 0 and solve for t. Once we find t, we can substitute it back into the equation to find the maximum altitude y.
4. Find the total time of flight:
The total time of flight is the time it takes for the rocket to reach its maximum altitude, plus the time it takes for the rocket to fall back to the ground. Since the maximum altitude occurs when vyi * t - (1/2) * g * t^2 = 0, we can use the same value of t to find the total time of flight.
5. Find the rocket's horizontal range:
The horizontal component of motion is not affected by the vertical acceleration. Therefore, the rocket's initial horizontal velocity and the acceleration do not play a role in finding the horizontal range. We can simply multiply the horizontal component of the initial velocity (vi * cos(è)) by the time of flight to find the horizontal range.
Let's calculate these values:
Given:
vi = 62 m/s
è = 48°
t = 25 s
g = 9.8 m/s^2
(a) What is the rocket's maximum altitude:
Calculate vyi = vi * sin(48°)
Plug in the values into the equation: vyi * t - (1/2) * g * t^2 = 0 and solve for t
Substitute the value of t into the equation y = y0 + vyi * t - (1/2) * g * t^2 to find y.
(b) What is the rocket's total time of flight:
The total time of flight is 2 * t because the rocket takes the same amount of time to reach its maximum altitude and fall back to the ground.
(c) What is the rocket's horizontal range:
Calculate the horizontal component of the initial velocity: vi * cos(48°)
Multiply this value by the total time of flight to find the horizontal range.
By following these steps and using the equations of motion, you can calculate and find the answers to the above questions.