A scientific calculator is required to answer the following question.
A particle [tex]$P$[/tex] is traveling along the [tex]$x$[/tex]-axis with displacement [tex]$s(t)$[/tex] from a fixed origin [tex]$O$[/tex] given by
[tex]$s(t)=(t+1) e^{-t}, \quad t \geq 0$[/tex]
Find the time at which the acceleration is zero, and find the value of the velocity at this moment.
a [tex]$t=2, \quad v=-\frac{2}{e^2}$[/tex]
b [tex]$t=1, \quad v=-\frac{1}{e}$[/tex]
c [tex]$t=\frac{1}{2}, \quad v=\frac{1}{2 \sqrt{e}}$[/tex]
d [tex]$t=\frac{1}{2}, \quad v=-\frac{1}{\sqrt{e}}$[/tex]
e [tex]$t=1, \quad v=\frac{1}{e^2}$[/tex]
Answer (2)
The acceleration of the particle is zero at t = 1 , and at this time, the velocity is v = − e 1 .
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Find the velocity function by differentiating the displacement function: v ( t ) = − t e − t .
Find the acceleration function by differentiating the velocity function: a ( t ) = e − t ( t − 1 ) .
Set the acceleration function to zero and solve for t : t = 1 .
Substitute t = 1 into the velocity function to find the velocity: v ( 1 ) = − e 1 .
t = 1 , v = − e 1
Explanation
Problem Setup We are given the displacement function s ( t ) = ( t + 1 ) e − t and asked to find the time when the acceleration is zero and the velocity at that time.
Finding the Velocity Function First, we need to find the velocity function v ( t ) , which is the derivative of the displacement function s ( t ) . Using the product rule, we have:
v ( t ) = s ′ ( t ) = d t d (( t + 1 ) e − t ) = ( 1 ) e − t + ( t + 1 ) ( − e − t ) = e − t − ( t + 1 ) e − t = e − t ( 1 − ( t + 1 )) = − t e − t
Finding the Acceleration Function Next, we need to find the acceleration function a ( t ) , which is the derivative of the velocity function v ( t ) . Using the product rule again, we have:
a ( t ) = v ′ ( t ) = d t d ( − t e − t ) = ( − 1 ) e − t + ( − t ) ( − e − t ) = − e − t + t e − t = e − t ( t − 1 )
Finding the Time When Acceleration is Zero Now, we need to find the time t when the acceleration is zero. Setting a ( t ) = 0 , we have:
e − t ( t − 1 ) = 0
Since e − t is never zero, we must have t − 1 = 0 , which gives t = 1 .
Finding the Velocity at that Time Finally, we need to find the velocity at t = 1 . Substituting t = 1 into the velocity function, we have:
v ( 1 ) = − 1 ⋅ e − 1 = − e 1
Final Answer Therefore, the time at which the acceleration is zero is t = 1 , and the velocity at that time is v = − e 1 .
Examples
Understanding the motion of particles is crucial in many fields. For instance, in physics, it helps analyze projectile motion or the behavior of objects under various forces. In engineering, it's used to design control systems for robots or vehicles, ensuring smooth and predictable movements. Even in economics, similar principles can model how markets respond to changes over time.
