Question

If a particle moves along a coordinate line so that its directed distance from the origin after $t$ seconds is $\left(-t^{2}+4 t\right)$ feet, when did the particle come to a momentary stop (i.e., when did its instantaneous velocity become zero)?

Name: Problem 16, Chapter 2: Calculus
Uploaded: 2021-06-13T12:11:10-08:00
Duration: 2 min 37 s
Channel: Chris Bolognese
Description: If a particle moves along a coordinate line so that its directed distance from the origin after t seconds is (-t^2+4 t) feet, when did the particle come to a momentary stop (i.e., when did its instantaneous velocity become zero)?

   If a particle moves along a coordinate line so that its directed distance from the origin after $t$ seconds is $\left(-t^{2}+4 t\right)$ feet, when did the particle come to a momentary stop (i.e., when did its instantaneous velocity become zero)?

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 2, Problem 16

Instant Answer

Solved by Expert Chris Bolognese

06/13/2021

Step 1

We need to find the time at which the particle comes to a momentary stop. This is equivalent to finding the time at which the velocity of the particle is zero. Show more…

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If a particle moves along a coordinate line so that its directed distance from the origin after $t$ seconds is $\left(-t^{2}+4 t\right)$ feet, when did the particle come to a momentary stop (i.e., when did its instantaneous velocity become zero)?

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Key Concepts

Position Function

The position function describes a particle’s location along a coordinate line as a function of time. It is fundamental in kinematics because it provides a way to track the particle's motion relative to a fixed reference point.

Derivative and Instantaneous Velocity

The derivative of the position function with respect to time is the instantaneous velocity. This concept from calculus quantifies the rate at which the particle's position changes at any given moment, making it essential for analyzing motion.

Setting the Derivative to Zero to Find Critical Points

Finding when the instantaneous velocity is zero involves setting the derivative of the position function equal to zero. Solving this equation identifies critical points in the motion where the particle momentarily stops, which is a key technique in determining changes in the particle’s motion.

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