Problem 102 - Thomas Calculus

Question

Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 \(\mathrm{m} / \mathrm{sec}^{2}\) . How fast will the rocket be going 1 min later?

Step-by-Step Solution

Answer

The rocket's velocity after 1 minute is 1200 m/s.

Step 1: Understanding the Problem

We need to calculate the final velocity of a rocket after 1 minute of constant acceleration. The rocket starts from rest, and its constant acceleration is given as 20 m/s².

Step 2: Identifying the Given Values

We recognize the following: Initial velocity (\( u \) = 0 m/s, since the rocket starts from rest), acceleration (\( a \) = 20 m/s²), and time (\( t \) = 1 min = 60 seconds) .

Step 3: Using the Right Formula

We use the equation of motion for velocity:\[ v = u + at \]where \( v \) = final velocity, \( u \) = initial velocity, \( a \) = acceleration, \( t \) = time.

Step 4: Substituting Values and Solving

Substitute the identified values into the equation:\[ v = 0 + 20 \times 60 \]\[ v = 1200 \text{ m/s} \]

Key Concepts

Initial Velocity

Initial velocity is the speed at which an object begins its motion. In this exercise, the rocket starts from rest, meaning its initial velocity is 0 m/s. This represents a situation where the rocket has not moved yet. The concept of initial velocity is crucial as it sets the starting point for equations of motion.

In scenarios involving constant acceleration, knowing the initial velocity helps us understand how much additional velocity will be added over time. If the initial velocity is not zero, it would be added to the change in velocity due to acceleration over time.

Commonly, the symbol used for initial velocity is \( u \). It serves as an anchor in calculations, ensuring that changes in speed over time have a reference point from which they are measured.

Final Velocity

Final velocity is the speed an object reaches after undergoing a specific amount of acceleration over a given period of time. It indicates how fast the object is moving at the end of the motion. In this exercise, the final velocity of the rocket is what we're solving for, and it results from the rocket's constant acceleration.

Final velocity is influenced by several factors:

• Initial Velocity: Where did the movement begin?
• Acceleration: How much does the velocity change per unit of time?
• Time: Over what duration does this change happen?

To calculate final velocity, we typically make use of one of the equations of motion, integrating the effects of initial velocity, acceleration, and time. In our problem, represented as \( v = u + at \), where \( v \) is the final velocity.

Equation of Motion

Equations of motion are mathematical formulas that describe the behavior of moving objects. These equations are pivotal in understanding and predicting motion under uniform acceleration, like in our exercise. The main equation relevant here is the velocity-time equation: \( v = u + at \).

This equation tells us about how velocity changes over time when acceleration is constant.

• \( u \): Initial velocity of the object.
• \( a \): Constant acceleration applied to the object.
• \( t \): Time duration over which the object is moving.

By substituting known values into this equation, we can solve for unknown quantities such as final velocity. This equation of motion is a fundamental tool in physics, enabling a precise analysis of how objects will move under specific conditions.