A test rocket is launched vertically from ground level (y = 0 m), at time t = 0.0 s. The rocket engine provides constant upward acceleration during the burn phase. At the instant of engine burnout, the rocket has risen to 49 m and acquired a velocity of What is the maximum height that the rocket will reach?
52.3
To determine the maximum height that the rocket will reach, we need to use the equations of motion for a vertically launched projectile.
First, we can determine the initial velocity of the rocket at the instant of engine burnout. Let's call this velocity "v0". We are given that the rocket has risen to a height of 49 m at this time. We also know that the initial velocity of the rocket was zero (since it started from rest). Therefore, the final velocity at burnout is also equal to v0.
Using the equation of motion for velocity in the vertical direction:
v = u + at
where:
v = final velocity (v0)
u = initial velocity (0 m/s)
a = acceleration
t = time
Since the rocket engine provides constant upward acceleration during the burn phase, we can assume that the acceleration remains constant until burnout. Therefore, we can rewrite the equation as:
v0 = 0 + a * t
Next, we can use the equation of motion for displacement in the vertical direction to determine the maximum height (h) reached by the rocket:
h = ut + (1/2) * a * t^2
Substituting the values we know:
h = 0 + (1/2) * a * t^2
Now, we need to find the value of time (t) at the instant of engine burnout. We are not given this information directly, but we do know that at this time, the rocket has risen to a height of 49 m. So, we can substitute this value into the equation and solve for t:
49 = (1/2) * a * t^2
Next, let's solve this equation for t by dividing both sides by (1/2) * a:
98/a = t^2
Taking the square root of both sides to solve for t:
t = sqrt(98/a)
Now, we have the value of time at engine burnout. We can substitute this into the equation for displacement to find the maximum height:
h = (1/2) * a * (sqrt(98/a))^2
h = (1/2) * a * (98/a)
h = 49 m
Therefore, the maximum height that the rocket will reach is 49 meters.