A toy rocket, launched from the ground, rises vertically with an acceleration of 23 m/s^2 for 11 s until its motor stops.
Disregarding any air resistance, what maximum height above the ground will the rocket achieve? The acceleration of gravity is 9.8 m/s^2.
Answer in units of k
To find the maximum height reached by the rocket, we can use the kinematic equation that relates displacement, initial velocity, final velocity, acceleration, and time.
The equation we will be using is:
s = ut + 0.5at^2
Where:
s = displacement (maximum height)
u = initial velocity
a = acceleration
t = time
First, we need to determine the initial velocity of the rocket. We know that the rocket starts from rest on the ground, so the initial velocity is 0 m/s.
Next, we need to find the time it takes for the rocket to reach its maximum height. Given that the rocket accelerates for 11 seconds, we can use this value for the time (t) in the equation.
Now, let's plug in the values into the equation:
s = (0 * 11) + (0.5 * 23 * 11^2)
Simplifying:
s = 0 + 0.5 * 23 * 121
s = 0 + 0.5 * 2803
s = 0 + 1401.5
s = 1401.5 m
The maximum height (displacement) reached by the rocket is 1401.5 meters above the ground.
To provide the answer in units of kilometers (k), we need to convert the meters to kilometers by dividing the result by 1000:
s = 1401.5 / 1000
s = 1.4015 km
Therefore, the maximum height reached by the rocket is approximately 1.4015 km above the ground.
To find the maximum height reached by the rocket, we need to use the formula:
h = (v^2 - u^2) / (2a)
where:
h = height
v = final velocity
u = initial velocity
a = acceleration
In this case, the rocket starts from rest (u = 0) and decelerates with an acceleration of -9.8 m/s^2 (opposite to the acceleration due to gravity) after the motor stops. The final velocity (v) at the maximum height is also 0 m/s.
We can divide the problem into two parts:
1. The rocket's ascent with an acceleration of 23 m/s^2.
2. The rocket's descent with an acceleration of -9.8 m/s^2.
1. Calculate the maximum height reached during ascent:
Using the formula, h = (v^2 - u^2) / (2a), we have:
h1 = (0 - 0^2) / (2 * 23)
h1 = 0 / 46
h1 = 0
2. Calculate the maximum height reached during descent:
Using the formula again, h = (v^2 - u^2) / (2a), we have:
h2 = (0 - 0^2) / (2 * -9.8)
h2 = 0 / -19.6
h2 = 0
The maximum height achieved by the rocket is the sum of the heights reached during ascent and descent:
h = h1 + h2
h = 0 + 0
h = 0
The rocket reaches a maximum height of 0 meters above the ground.