The acceleration of a particle traveling along a straight line is #a = (8-2s) m/s^s#, where #s# is in meters. How do you determine the position of the particle when the velocity is maximum?

Question

The acceleration of a particle traveling along a straight line is #a = (8-2s) m/s^s#, where #s# is in meters.  How do you determine the position of the particle when the velocity is maximum?

Lincoln Adams · Accepted Answer

I got at #4m#

I would use Algebra and Kinematics here : I know that acceleration is the derivative of velocity BUT also that if I want the maximum of a function I need to derive it and set the derivative equal to zero; here I have the derivative of velocity (acceleration) so I set it equal to zero to get: #8-2s=0# #s=8/2=4m#

Does that make sense?

Theodore Angelis · Answer

undefined In order to find the particle's maximum velocity position, we first find the velocity function by integrating the acceleration function with respect to time. Next, we solve for the position corresponding to these critical points by setting the derivative of the velocity function equal to zero. Lastly, we use the second derivative test to find out if these critical points correspond to maximum velocity.

Leah Angelini · Answer

undefined To determine the position of the particle when the velocity is maximum, you first find the velocity function by integrating the acceleration function with respect to time. Then, you identify critical points of the velocity function by setting its derivative equal to zero and solving for $ s $. Finally, you evaluate the position function at these critical points to find the position when the velocity is maximum.