Tadeh launches a model rocket straight up from his backyard that takes 6.96 s to reach its maximum altitude. (After launch, the rocket's motion is only influenced by gravity. Enter magnitudes only.)
(a) What is the rocket's initial velocity?
m/s
(b) What is the maximum altitude reached by the rocket?
m
a)
Use the equation of motion: v = u + at
=> v = 0 m/s, a = -9.8 m/s^2, t = 6.96 sec
=> u = v - at = 0 - (-9.8)(6.96)
= 68.2 m/s
b)
Use the equation of motion: 2as = v^2 - u^2
=> a = -9.8 m/s^2, v = 0, u = 68.2 m/s
=> 2(-9.8)h = -(68.2)^2
=> 2*h = (68.2)(6.96)
=> h = 474/2
= 237 m
To determine the initial velocity of the rocket and the maximum altitude reached, we can use the following kinematic equation:
v = u + at
Where:
v = final velocity (which is 0 m/s at the maximum altitude)
u = initial velocity
a = acceleration (which is equal to the acceleration due to gravity, approximately 9.8 m/s^2)
t = time taken to reach the maximum altitude (6.96 s)
(a) To find the initial velocity (u), we can rearrange the equation:
u = v - at
Since the final velocity (v) is 0 m/s, the equation becomes:
u = -at
Substituting the values:
u = -9.8 m/s^2 * 6.96 s = -68.208 m/s ≈ -68.2 m/s
Therefore, the initial velocity of the rocket is approximately -68.2 m/s.
(b) To find the maximum altitude reached, we need to use the equation for displacement:
s = ut + (1/2)at^2
Since the final displacement (s) is 0 m (as the rocket reaches maximum altitude and starts descending), the equation becomes:
0 = ut + (1/2)at^2
We can rearrange the equation to solve for the maximum altitude (u):
u = - (1/2)at^2 / t
Substituting the values:
u = - (1/2) * 9.8 m/s^2 * (6.96 s)^2 / 6.96 s = -240.596 m ≈ -240.6 m
Therefore, the maximum altitude reached by the rocket is approximately -240.6 m. Please note that the negative sign indicates that the rocket is moving in the opposite direction of the gravitational field.
To find the initial velocity, we can use the equation of motion:
𝑣 = 𝑢 + 𝑎𝑡
Where:
- 𝑣 is the final velocity (which is zero when the rocket reaches its maximum altitude)
- 𝑢 is the initial velocity (what we're trying to find)
- 𝑎 is the acceleration due to gravity (-9.8 m/s² for objects near the surface of the Earth)
- 𝑡 is the time taken (6.96 s in this case)
(a) To find the initial velocity:
0 = 𝑢 + (-9.8 m/s²) * 6.96 s
We can rearrange the equation to solve for 𝑢:
𝑢 = 9.8 m/s² * 6.96 s
= 68.208 m/s
So the rocket's initial velocity is 68.208 m/s.
To find the maximum altitude reached by the rocket, we can use the equation of motion:
𝑠 = 𝑢𝑡 + 0.5𝑎𝑡²
Where:
- 𝑠 is the maximum altitude (what we're trying to find)
- 𝑢 is the initial velocity (which we just calculated)
- 𝑡 is the time taken (6.96 s in this case)
- 𝑎 is the acceleration due to gravity (-9.8 m/s²)
(b) To find the maximum altitude:
𝑠 = (68.208 m/s) * 6.96 s + 0.5 * (-9.8 m/s²) * (6.96 s)²
Simplifying the equation:
𝑠 = 474.6144 m + 0.5 * (-9.8 m/s²) * 48.4416 s²
= 474.6144 m - 237.3072 m
= 237.3072 m
So the rocket's maximum altitude is 237.3072 m.