Problem 5 - Calculus Early Transcendentals

Question

Define the acceleration of an object moving in a straight line.

Step-by-Step Solution

Answer

Answer: The acceleration of an object moving in a straight line is the rate at which the object's velocity changes over time, along a single dimension, taking into account both the magnitude and direction of these changes. It is mathematically represented as: a = \frac{\Delta v}{\Delta t} where a is the acceleration, Δv is the change in velocity, and Δt is the change in time.

Step 1: Understand the concept of acceleration

Acceleration is a measure of how quickly an object's velocity changes over a period of time. It is a vector quantity, which means it has both a magnitude (how much the velocity changes) and a direction (whether the object is speeding up or slowing down).

Step 2: Define velocity

Velocity is the rate at which an object changes its position. It is also a vector quantity, with magnitude (speed) and direction. Mathematically, velocity (v) can be represented as the change in position (Δx) divided by the change in time (Δt), or: v = \frac{\Delta x}{\Delta t}

Step 3: Define acceleration

Acceleration (a) is the rate at which an object's velocity changes over time. Like velocity, it is also a vector quantity. Mathematically, acceleration can be represented as the change in velocity (Δv) divided by the change in time (Δt), or: a = \frac{\Delta v}{\Delta t}

Step 4: Illustrate an object moving in a straight line

When an object moves in a straight line, its motion is one-dimensional. This means its acceleration is also one-dimensional and only has one component, which can be either positive or negative. A positive acceleration means the object is speeding up, while a negative acceleration (also called deceleration) means it is slowing down.

Step 5: Define the acceleration of an object moving in a straight line

Based on the understanding of the acceleration and one-dimensional motion of an object, we can define the acceleration of an object moving in a straight line as: The rate at which the object's velocity changes over time, along a single dimension, taking into account both the magnitude and direction of these changes. Mathematically, it is represented as: a = \frac{\Delta v}{\Delta t} where a is the acceleration, Δv is the change in velocity, and Δt is the change in time.