Under what condition is average velocity equal to instantaneous velocity?
Answer (2)
Instantaneous velocity is defined as the derivative of the position function: v ( t ) = d t d x ( t ) .
Average velocity over an interval [ t 1 , t 2 ] is given by v a vg = t 2 − t 1 x ( t 2 ) − x ( t 1 ) .
According to the Mean Value Theorem, if x ( t ) is continuous on [ t 1 , t 2 ] and differentiable on ( t 1 , t 2 ) , there exists a c in ( t 1 , t 2 ) such that x ′ ( c ) = t 2 − t 1 x ( t 2 ) − x ( t 1 ) .
Thus, instantaneous velocity equals average velocity at some time t = c within the interval ( t 1 , t 2 ) , i.e., v ( c ) = v a vg , provided the conditions of the Mean Value Theorem are satisfied. v ( c ) = v a vg
Explanation
Problem Statement We are asked to find the condition under which instantaneous velocity is equal to average velocity.
Instantaneous Velocity Let's define instantaneous velocity as v ( t ) = d t d x ( t ) , where x ( t ) is the position function with respect to time t .
Average Velocity The average velocity over the interval [ t 1 , t 2 ] is defined as v a vg = t 2 − t 1 x ( t 2 ) − x ( t 1 ) , where x ( t 2 ) and x ( t 1 ) are the positions at times t 2 and t 1 , respectively.
Equating Velocities We want to find the condition when the instantaneous velocity equals the average velocity, i.e., when v ( t ) = v a vg . This means we want to find when d t d x ( t ) = t 2 − t 1 x ( t 2 ) − x ( t 1 ) .
Applying the Mean Value Theorem The Mean Value Theorem provides the condition we are looking for. The Mean Value Theorem states that if a function x ( t ) is continuous on the closed interval [ t 1 , t 2 ] and differentiable on the open interval ( t 1 , t 2 ) , then there exists a point c in ( t 1 , t 2 ) such that x ′ ( c ) = t 2 − t 1 x ( t 2 ) − x ( t 1 ) . In our context, x ′ ( c ) is the instantaneous velocity v ( c ) at time t = c .
Conclusion Therefore, the instantaneous velocity equals the average velocity when the conditions of the Mean Value Theorem are met. Specifically, there must exist a time t = c in the interval ( t 1 , t 2 ) such that v ( c ) = t 2 − t 1 x ( t 2 ) − x ( t 1 ) .
Examples
Imagine you're tracking a race car's speed. The average speed over a lap is easy to calculate, but the instantaneous speed is what the car's speedometer reads at any given moment. The Mean Value Theorem tells us that at some point during the lap, the car's instantaneous speed had to match its average speed for that lap. This concept is crucial in physics and engineering for analyzing motion and performance over time.
Average velocity equals instantaneous velocity when certain conditions are met, specifically when the position function is continuous and differentiable over a given time interval, per the Mean Value Theorem. This theorem guarantees that there is at least one moment in the interval where both velocities match. Thus, if an object moves smoothly without sudden changes, average and instantaneous velocity will coincide at some point.
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