A model rocket is launched straight up from ground level with a speed of 34.2 m/s.
Assuming that the only force acting on the rocket after launch is the force of gravity,
a. How fast is the rocket going when it reaches a height of 22.5 m? (3 marks)
b. How long does it take the rocket to reach this height? (2 marks)
c. Why are there two answers to part b? Explain with a sketch. (1 mark
a. To calculate the speed of the rocket when it reaches a height of 22.5 m, we can use the equation for final velocity (v) given initial velocity (u), acceleration (a), and displacement (s):
v^2 = u^2 + 2as
Since the rocket is launched straight up, the acceleration due to gravity (a) is acting downward, which we can take as -9.8 m/s^2. The initial velocity (u) is 34.2 m/s and the displacement (s) is 22.5 m. Plugging these values into the equation, we can solve for the final velocity (v).
v^2 = 34.2^2 + 2(-9.8)(22.5)
v^2 = 1169.64 - 441
v^2 = 728.64
v = 26.98 m/s (rounded to two decimal places)
Therefore, the rocket is going approximately 26.98 m/s when it reaches a height of 22.5 m.
b. To calculate the time it takes for the rocket to reach a height of 22.5 m, we can use the equation for displacement (s) given initial velocity (u), time (t), and acceleration (a):
s = ut + (1/2)at^2
Rearranging the equation to solve for time (t):
t = sqrt((2s) / a)
Plugging in the values of displacement (s = 22.5 m) and acceleration due to gravity (a = -9.8 m/s^2), we can find the time it takes for the rocket to reach 22.5 m.
t = sqrt((2 * 22.5) / -9.8)
t = sqrt(-45.9 / -9.8)
t = sqrt(4.6897959)
t = 2.165 s (rounded to three decimal places)
Therefore, it takes approximately 2.165 seconds for the rocket to reach a height of 22.5 m.
c. There are two answers to part b because the rocket reaches the height of 22.5 m twice during its trajectory, once while going up and once while coming down. The first time the rocket reaches this height is during its ascent, and the second time is during its descent.
To better understand this, we can look at a sketch of the rocket's trajectory. The vertical axis represents the displacement (height) of the rocket, and the horizontal axis represents time.
/|
/ |
/ |
/ |
/ | (second instance of 22.5 m)
/ |
/______|
| /
| /
| /
| /
| /
|/
(first instance of 22.5 m)
As shown in the sketch, the rocket first reaches a height of 22.5 m during its ascent, and then again during its descent. The time it takes to reach this height is the same for both instances, which is approximately 2.165 seconds as calculated in part b.