A rocket is launched at an angle of 58.0° above the horizontal with an initial speed of 97 m/s. The rocket mov

Question

A rocket is launched at an angle of 58.0° above the horizontal with an initial speed of 97 m/s. The rocket moves for 3.00 s along its initial line of motion with an acceleration of 28.0 m/s2. At this time, its engines fail and the rocket proceeds to move as a projectile.
(a) Find the maximum altitude reached by the rocket.
m
(b) Find its total time of flight.
s
(c) Find its horizontal range.
m

Answer 1

This is piecewise linear
I made an assumption that 28 m/s^2 is the net acceleration along a straight 58 degree flight path.

0 For the first part of the motion it has vertical velocity as

vy(t)=sin(58)*(97+28*t)
and
y(t)=sin(58)*(97*t+14*t^2)

for t>3 until y(t)=0
we must calculate the speed at the end of the first 3 s
153.5 m/s
The altitude at this moment is
353.6 meters
so the equations for t>3 are
vy(t)=153.5-g*(t-3)
y(t)=353.6+153.5*(t-3)-.5*g*(t-3)^2
using g=9.81
at apogee, vy(t)=0
0=153.5-g*(t-3)
t=153.5/g+3
t=18.65 seconds

so the altitude is
1554 meters

the total flight time is found by setting y(t)=0 and solving the quadratic for the larger root

t=36.45 seconds this is the total flight time.

It is always a good idea to reconcile the existence of a second root. The second root is 0.84 seconds. This value is the equivalent time that if the rocket were given a vy0 from the ground that would bring it to apogee at the time and altitude that the rocket motor did. The root is positive since the acceleration of the rocket motor was almost 3 times g, and the angle was great enough at 58 degrees.

For range
0 vx(t)=cos(58)*(97+28*t)
x(t)=cos(58)*(97*t+14*t^2)
the horizontal speed at t=3 is

95.9 m/s
and the distance down range is
221 meters

for 3 x(t)=221+95.9*(t-3)
from above we know that impact occurs at 36.45 seconds
x(36.45)=3429 meters

j

Answer 2

take the angle and also the speed of the rocket from there work out the loco of where the rocket will land like a compass taking the loco of the circle but the speed progress as the rocket drops