Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 56a

Chapter 2, Problem 56a

A 200 kg weather rocket is loaded with 100 kg of fuel and fired straight up. It accelerates upward at 30 m/s² for 30 s, then runs out of fuel. Ignore any air resistance effects. What is the rocket's maximum altitude?

Verified step by step guidance

Step 1: Calculate the velocity of the rocket at the end of the fuel burn phase using the formula for final velocity under constant acceleration:

v = u + a t

, where

u

is the initial velocity (0 m/s),

a

is the acceleration (30 m/s²), and

t

is the time (30 s).

Step 2: Calculate the altitude reached during the fuel burn phase using the kinematic equation:

s = u t + \frac{1}{2} a t ²

, where

u

is the initial velocity (0 m/s),

a

is the acceleration (30 m/s²), and

t

is the time (30 s).

Step 3: Determine the rocket's motion after the fuel runs out. The rocket will continue to ascend due to its velocity at the end of the fuel burn phase. Use the kinematic equation

s = \frac{v}{2} / g

to calculate the additional altitude gained during this phase, where

v

is the velocity at the end of the fuel burn phase and

g

is the acceleration due to gravity (9.8 m/s²).

Step 4: Add the altitude from the fuel burn phase to the altitude gained during the coasting phase to find the total maximum altitude of the rocket.

Step 5: Ensure all units are consistent throughout the calculations (e.g., meters, seconds) and verify the results conceptually to confirm the rocket's maximum altitude.

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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. This principle is crucial for understanding how the rocket accelerates upward when the engines are firing, as the net force is the difference between the thrust produced by the rocket and the gravitational force acting on it.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. In this scenario, they can be used to calculate the rocket's velocity and displacement during the powered ascent phase, as well as the subsequent free-fall phase after the fuel runs out, allowing us to determine the maximum altitude reached.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. As the rocket ascends, it converts kinetic energy into gravitational potential energy. Understanding this concept is essential for calculating the maximum altitude, as it involves determining the point where the rocket's kinetic energy is fully converted into potential energy before it starts descending.

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