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Chapter 2: Problem 1

Under what conditions are average and instantaneous velocity equal?

Short Answer

Expert verified
The average and instantaneous velocities are equal when the velocity is constant, i.e. the object is moving at a constant speed without acceleration.

Step by step solution

01

Define Average Velocity

The average velocity is calculated by taking the total displacement (change in position) and dividing it by the total time it takes for the displacement. The formula is \( v_{avg} = \frac{\Delta x}{\Delta t} \) where \( \Delta x \) is the change in position and \( \Delta t \) is the change in time.
02

Define Instantaneous Velocity

The instantaneous velocity is the velocity of an object at a specific point in time. It is calculated as the limit of the average velocity as the time interval approaches zero. Formally, the formula is \( v_{inst} = \lim_{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t} \) . This gives the 'instantaneous' rate of change of position.
03

Equate the velocities

For average and instantaneous velocities to be equal, the velocity of the object must be constant, meaning it doesn't change over time. This means there is no acceleration going on, the object's speed doesn't increase or decrease. Therefore, whether looking at the average over a period of time, or at any specific moment in time, the velocity must be the same.

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