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?

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To find the highest altitude reached by the rocket, we need to determine the time it takes to reach the highest point and then calculate the vertical distance traveled during that time.

Step 1: Find the time taken to reach the highest point.
Since the rocket is initially moving along its initial line of motion, the time taken to reach the highest point will be half of the total time of flight. So, we first need to find the total time of flight.

Step 2: Find the total time of flight.
To find the total time of flight, we can use the equation of motion:
h = v₀t + 0.5at²

Where:
h = vertical displacement (highest altitude reached)
v₀ = initial vertical velocity (in this case, 0, as the rocket is launched horizontally)
t = time of flight
a = vertical acceleration (in this case, acceleration due to gravity, which is approximately -9.8 m/s² for the rocket moving upward)

However, initially, the rocket moves for 3.00 seconds with an acceleration of 29.0 m/s² along its initial line of motion. So, we need to subtract this time from the total time of flight.

Step 3: Find the total time of flight.
Using the equation of motion and substituting the given values, we get:
0 = (0)(t) + 0.5(-9.8)(t)²

Simplifying and solving this quadratic equation will give us the total time of flight.

Step 4: Find the highest altitude reached.
Once we have the total time of flight, we can substitute this value into the equation of motion to find the highest altitude reached:
h = v₀t + 0.5at²

In this case, the initial vertical velocity is 0, so the equation simplifies to:
h = 0.5at²

Substituting the known values, we can calculate the highest altitude reached by the rocket.

To find the horizontal range, we can use the equation of motion in the horizontal direction.

Step 5: Find the horizontal range.
The horizontal velocity of the rocket remains constant throughout its motion, as there are no horizontal forces acting on it. The velocity in the horizontal direction can be calculated using the initial speed and the angle of launch.

vx = v₀x = v₀ * cos(θ)

Where:
vx = horizontal velocity
v₀x = initial horizontal velocity
v₀ = initial speed of the rocket
θ = angle of launch

Now, the horizontal range can be calculated using the equation:
range = vx * time of flight

Substituting the known values, we can calculate the horizontal range.

By following these steps, you can find the highest altitude reached by the rocket, the total time of flight, and the horizontal range.