A model rocket blasts off from the ground, rising straight upward with a constant acceleration that has a magnitude of 88.6 m/s2 for 1.57 seconds, at which point its fuel abruptly runs out. Air resistance has no effect on its flight. What maximum altitude (above the ground) will the rocket reach?
Dear student,
ok when the fuel runs out, acceleration = 0.. however the rocket is still going up until the forces of gravity finally make it change direction.
so we must find 2 distances, one from launchpad till fuel runs out, then from the point where fuel runs out to our highest altitude.
a = 86 m/s^2
t = 1.7 seconds
Step 1: Find the distance to the point where fuel runs out
d = vi(t) + .5(a)(t^2)
d = (0)(1.7) + .5(86)(1.7)^2
d1 = 124.27m
Step 2: Find the velocity at the point where the fuel runs out
vf = vi + at
vf = 0 + 86(1.7)
vf = 146.2 m/s (this is our velocity when the fuel runs out)
Step 3: Find the time of our new distance equation (vf = 0 = maximum altitude)
(hint: at this point, gravity kicks in because the rocket stops accelerating)
vf = vi + at
0 = 146.2 + (-9.8)(t)
t = 14.92
Step 4: Find the distance up until the point where vf = 0 or t =15 seconds (maximum altitude before the rocket switches direction)
d= vi(t) + .5(a)(t^2)
d = 146.2(14.92) + .5(-9.8)(14.92^2)
d2 = 1 090.53
Step 5: Add our distances to find maximum altitude:
so our total distance above the ground is d1 + d2 =
124.27m + 1090.53m = 1 214.8 m
Eevee,
To find the maximum altitude reached by the rocket, we need to determine the displacement during the first 1.57 seconds of its flight.
We can use the kinematic equation that relates displacement, initial velocity, time, and acceleration:
d = v_i * t + (0.5) * a * t^2
Where:
d = displacement
v_i = initial velocity
t = time
a = acceleration
In this case:
v_i = 0 m/s (since the rocket starts from rest)
t = 1.57 seconds
a = 88.6 m/s^2
Substituting these values into the equation:
d = 0 * 1.57 + (0.5) * 88.6 * (1.57)^2
Simplifying:
d = 0 + (0.5) * 88.6 * 2.4649
d = 108.93 meters
Therefore, the maximum altitude reached by the rocket is 108.93 meters above the ground.
To find the maximum altitude reached by the rocket, we can use the kinematic equation:
h = h0 + V0t + (1/2)at²
Where:
h = maximum altitude
h0 = initial height (which is 0 since the rocket starts from the ground)
V0 = initial velocity (which is also 0 since the rocket starts from rest)
t = time
a = acceleration
Given:
a = 88.6 m/s²
t = 1.57 s
Using these values, we can calculate the maximum altitude reached by substituting the values into the equation:
h = 0 + 0 × 1.57 + (1/2) × 88.6 × (1.57)²
Simplifying the equation:
h = (1/2) × 88.6 × 2.4649
h = 109.3283 meters
Therefore, the maximum altitude reached by the rocket is approximately 109.3283 meters above the ground.