A 75007500-kg rocket blasts off vertically from the launch pad with a constant upward acceleration of 2.252.25 m/s2 and feels no appreciable air resistance. When it has reached a height of 525525 m, its engines suddenly fail; the only force acting on it is now gravity. What is the maximum height this rocket will reach above the launch pad?

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 02: Motion Along a Straight Line

Problem 43a

Chapter 2, Problem 43a

A $7500$ -kg rocket blasts off vertically from the launch pad with a constant upward acceleration of $2.25$ m/s² and feels no appreciable air resistance. When it has reached a height of $525$ m, its engines suddenly fail; the only force acting on it is now gravity. What is the maximum height this rocket will reach above the launch pad?

Verified step by step guidance

First, calculate the initial velocity of the rocket at the moment the engines fail using the kinematic equation:

v_{f} = v_{i} + a t

. Since the rocket starts from rest,

v_{i}

is 0, and you can use the equation

v_{f} = a t

Next, determine the time it takes for the rocket to reach the height of 525 m using the equation:

s_{f} = s_{i} + v_{i} t + \frac{1}{2} a t^{2}

. Here,

s_{i}

is 0, and

v_{i}

is 0, so the equation simplifies to

s_{f} = \frac{1}{2} a t^{2}

. Solve for

t

Once you have the time, substitute it back into the equation for final velocity to find the velocity at the height of 525 m:

v_{f} = a t

Now, calculate the maximum height the rocket will reach after the engines fail using the equation for motion under gravity:

{v_{f}}^{2} = {v_{i}}^{2} - 2 g s_{f}

, where

g

is the acceleration due to gravity (9.81 m/s²). Set

v_{f}

to 0 to find the maximum height.

Finally, add the height reached before the engines failed (525 m) to the additional height calculated in the previous step to find the total maximum height above the launch pad.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this scenario, the rocket's upward acceleration is due to the thrust provided by its engines, which can be calculated using the formula F = ma, where F is the force, m is the mass, and a is the acceleration.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They are essential for calculating the rocket's velocity and position at any given time. In this problem, these equations help determine the rocket's velocity at the moment the engines fail and predict its subsequent motion under the influence of gravity alone.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transformed. As the rocket ascends, its kinetic energy is converted into potential energy. After the engines fail, the rocket's maximum height can be found by equating its initial kinetic energy at engine failure to the potential energy at its peak height, considering gravity as the only acting force.

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