A rocket is launched at an angle of 56.0° above the horizontal with an initial speed of 101 m/s. The rocket moves for 3.00 s along its initial line of motion with an acceleration of 29.0 m/s2. At this time, its engines fail and the rocket proceeds to move as a projectile.

What is the highest altitude reached by the rocket?
Total time of flight?
Horizontal range?

break it into two parts, solve each independently, the acceleration part, the free fall part. Then add the distances.

To find the highest altitude reached by the rocket, total time of flight, and horizontal range, we can use the equations of motion for projectile motion.

1. Highest Altitude Reached:
The rocket reaches its highest altitude at the peak of its trajectory when its vertical velocity becomes zero. We can find this using the equation for vertical displacement:
Δy = Vyi * t - 1/2 * g * t^2
where Δy is the vertical displacement, Vyi is the initial vertical velocity, t is the time, and g is the acceleration due to gravity (-9.8 m/s^2).

Given:
Vyi = V * sin(θ), where V is the initial speed of the rocket and θ is the launch angle.
t = time taken for the engines to fail = 3.00 s

Substituting the given values:
Δy = (V * sin(θ)) * t - 1/2 * g * t^2

2. Total Time of Flight:
The total time of flight is the time taken for the rocket to reach the ground. We can find this using the equation for vertical motion:
Vyi = Vi + g * t
where Vi is the final vertical velocity, given as zero since the rocket reaches the ground.

Substituting the values:
0 = (V * sin(θ)) - g * t

Rearranging the equation:
t = (V * sin(θ)) / g

The total time of flight is twice the time taken for the engines to fail.

3. Horizontal Range:
The horizontal range is the total horizontal distance covered by the rocket. We can find this using the equation for horizontal displacement:
Δx = Vxi * t
where Δx is the horizontal displacement, Vxi is the initial horizontal velocity, and t is the total time of flight.

Given:
Vxi = V * cos(θ)

Substituting the values:
Δx = (V * cos(θ)) * t

Now, we can substitute the given values into the equations and calculate the results.