91Ó°ÊÓ

Launch Failure. \(A 7500\) -kg rocket blasts off vertically from the launch pad with a constant upward acceleration of 2.25 \(\mathrm{m} / \mathrm{s}^{2}\) and feels no appreciable air resistance. When it has reached a height of \(525 \mathrm{m},\) its engines suddenly fail so that the only force acting on it is now gravity. (a) What is the maximum height this rocket will reach above the launch pad? (b) How much time after engine failure will elapse before the rocket comes crashing down to the launch pad, and how fast will it be moving just before it crashes? (c) Sketch \(a_{y^{-}} t, v_{y^{-}} t,\) and \(y-t\) graphs of the rocket's motion from the instant of blast-off to the instant just before it strikes the launch pad.

Step by step solution

01

Analyze Rocket's Launch Motion

First, determine the rocket's velocity at engine failure using the equation: \(v = u + at\). Here, \(u = 0\) (initial velocity), \(a = 2.25 \, \mathrm{m/s^2}\) (acceleration), and \(s = 525 \, \mathrm{m}\) (height). Use \(v^2 = u^2 + 2as\) to find the velocity: \(v^2 = 0 + 2 \cdot 2.25 \cdot 525\). Calculate \(v\).

02

Determine Maximum Height

After the engine fails, the rocket continues upward due to its inertia until its velocity reaches zero. Use \(v^2 = u^2 + 2as\), where \(v = 0\) and \(a = -9.81 \, \mathrm{m/s^2}\) (gravity). Solve for \(s\) to find the additional height reached beyond 525 m. Add this to the initial 525 m to find the maximum height.

03

Calculate Time of Descent

When the rocket reaches maximum height, it starts coming down. Use \(v = u + at\) to calculate the time it takes to fall back to the launch pad. With \(u = 0\) at the top, find total time from max height using \(y = ut + \frac{1}{2}at^2\) set to the max height obtained.

04

Compute Impact Velocity

Use \(v^2 = u^2 + 2as\) with \(u = 0\) and \(a = 9.81 \, \mathrm{m/s^2}\) for total descent to calculate the velocity just before impact.

05

Draw Motion Graphs

Create three graphs: 1) Acceleration vs. time: starts constant at 2.25 m/s², drops to -9.81 m/s². 2) Velocity vs. time: increases linearly to max, decreases to zero at peak, and increases negatively. 3) Position vs. time: quadratic increase, maximum at peak height, then quadratic decrease back to zero at the pad.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects. It focuses on parameters like position, velocity, and acceleration without considering the forces that cause this motion. In this rocket launch scenario, kinematics helps us to describe how the rocket moves upward and downward.
To understand this, you would use equations of motion to describe:

• Initial motion upwards with acceleration provided by the rocket engines.
• Continuous rise until the maximum height after engine failure, driven solely by inertia.
• Descent back to the ground under the influence of gravity.

The key idea is that these motions are underpinned by different types of kinematic behaviors linked to changing velocities and accelerations.

Acceleration

Acceleration describes how an object's velocity changes over time. In the case of the rocket, it initially has a constant acceleration provided by its engines, which is 2.25 m/s² upward.
Upon engine failure, the only acceleration acting on the rocket is due to gravity, which is -9.81 m/s² downward. This shift signifies a transition from powered ascent to free fall. The constant positive acceleration allows the rocket to gain speed as it climbs. Once the engines fail, the negative acceleration due to gravity will gradually slow the rocket until it reaches its peak. After reaching maximum height, gravity continues to act, causing the rocket to accelerate downwards towards the launch pad.

Velocity

Velocity refers to the speed of an object in a specific direction. Initially, the rocket's velocity increases from rest (0 m/s) due to its upward acceleration.
Using the kinematic equation \( v^2 = u^2 + 2as \), the velocity just before engine failure can be calculated, where \( u \) is the initial velocity.When the engines fail, the rocket continues to climb until its velocity becomes 0 m/s at the maximum height. Afterwards, the velocity turns negative as the rocket starts descending. By descending, it's picking up speed in the downward direction, under the constant acceleration of gravity.

Free Fall

Free fall occurs when the only force acting on an object is gravity. After the rocket’s engines fail, it enters a free fall state.
In this state, the previous upward motion slows to a stop before the rocket starts descending due to gravity. An important aspect to note during free fall is that the downward acceleration remains constant at -9.81 m/s², irrespective of the upward motion or the descending phase. This helps determine both the maximum height reached after ascent and how fast the rocket will be moving as it crashes back to the launch pad.

Graphs of Motion

Graphs are essential tools in physics to visually interpret kinematic concepts. Three main graphs can describe the rocket's motion:

• Acceleration vs. Time: Begins with a constant 2.25 m/s² during the powered ascent, then drops to -9.81 m/s² after engine failure, indicating free fall.
• Velocity vs. Time: Shows a linear increase in velocity during powered ascent, which becomes zero at the maximum height. As the rocket falls, velocity increases negatively.
• Position vs. Time: Displays a quadratic increase in the rocket's height during ascent, peaks when the maximum height is reached, then shows a symmetric quadratic decrease during descent back to the launch pad.

These graphs provide insight into how different phases of the motion are interrelated and how parameters change over time.