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Chapter 1: Problem 15

What is the relationship between velocity and acceleration?

Short Answer

Expert verified
Answer: The relationship between velocity and acceleration is that acceleration is the rate of change of velocity with respect to time. Positive acceleration indicates an increasing velocity, while negative acceleration (deceleration) indicates a decreasing velocity. If acceleration is zero, the velocity remains constant.

Step by step solution

01

1. Defining Velocity

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. Mathematically, it can be represented as the first derivative of position with respect to time: v(t) = \frac{d}{dt} x(t), where v(t) is the velocity at time t and x(t) is the position function.
02

2. Defining Acceleration

Acceleration is also a vector quantity that represents the rate of change of an object's velocity with respect to time. It can be expressed as the first derivative of velocity or the second derivative of position with respect to time: a(t) = \frac{d}{dt} v(t) = \frac{d^2}{dt^2} x(t), where a(t) is the acceleration at time t.
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3. Explaining the Relationship Between Velocity and Acceleration

The relationship between velocity and acceleration can be best understood through their mathematical representations. Since acceleration is the derivative of velocity, it describes how velocity is changing over time. Positive acceleration indicates that the velocity is increasing, whereas negative acceleration (also called deceleration) indicates that the velocity is decreasing. When acceleration is zero, the velocity is constant, meaning the object is either not moving or it's moving at a constant speed. In summary, acceleration is the rate of change of velocity, which allows us to understand how an object's velocity is changing over time.

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