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Chapter 2: Problem 1

Under what conditions are average and instantaneous velocity equal?

Short Answer

Expert verified
Average and instantaneous velocities are equal under the condition of uniform or constant velocity, where the object covers equal distances in equal time intervals.

Step by step solution

01

Define average velocity

Average velocity is defined as the total displacement (the final position minus the initial position) divided by the total time. It's mathematically expressed as \( V_{avg} = \frac{X_{f} - X_{i}}{t_{f} - t_{i}} \) where \( X_{f} \) and \( X_{i} \) are the final and initial positions, respectively, and \( t_{f} \) and \( t_{i} \) are the final and initial times, respectively.
02

Define instantaneous velocity

Instantaneous velocity is the velocity of an object at a specific instant. When the time interval \( \Delta t \) approaches 0, the average velocity becomes the instantaneous velocity. It can be mathematically expressed as \( V_{ins} = \lim_{\Delta t\to 0} \frac{\Delta x}{\Delta t} \) where \( \Delta x \) is the small displacement in position and \( \Delta t \) is the small interval of time.
03

Determine the conditions

Average and instantaneous velocities are equal when the velocity of the object doesn't change throughout the time interval i.e., when the motion is uniform. In such a scenario, the object covers equal distances in equal time intervals, and hence, there is no difference between the average and instantaneous velocities.

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