A particle is moving along the [tex]$x$[/tex] axis such that its position is given by [tex]$x=4 t^2-16 t+12$[/tex].

i. Find the instantaneous velocity when [tex]$t=5 s$[/tex].
ii. Find the instantaneous acceleration when [tex]$t =5 s$[/tex].
iii. At what time is the particle stationary?

The instantaneous velocity at t = 5 s is 24 m/s, the instantaneous acceleration is 8 m/s², and the particle is stationary at t = 2 s.
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Answered by Anonymous | 2025-07-04

Find the velocity function by taking the derivative of the position function: v ( t ) = 8 t − 16 .
Find the acceleration function by taking the derivative of the velocity function: a ( t ) = 8 .
Calculate the instantaneous velocity at t = 5 s: v ( 5 ) = 24 .
Determine the time when the particle is stationary by setting v ( t ) = 0 , which gives t = 2 .

v ( 5 ) = 24 , a ( 5 ) = 8 , t = 2 ​
Explanation

Problem Setup We are given the position function of a particle moving along the x-axis as x ( t ) = 4 t 2 − 16 t + 12 . We need to find the instantaneous velocity at t = 5 s, the instantaneous acceleration at t = 5 s, and the time when the particle is stationary.

Finding the Velocity Function To find the instantaneous velocity, we need to find the derivative of the position function with respect to time, which gives us the velocity function v ( t ) .


v ( t ) = d t d x ​ = d t d ​ ( 4 t 2 − 16 t + 12 ) = 8 t − 16

Finding the Acceleration Function To find the instantaneous acceleration, we need to find the derivative of the velocity function with respect to time, which gives us the acceleration function a ( t ) .

a ( t ) = d t d v ​ = d t d ​ ( 8 t − 16 ) = 8

Calculating Velocity at t=5s Now, we can find the instantaneous velocity at t = 5 s by substituting t = 5 into the velocity function.

v ( 5 ) = 8 ( 5 ) − 16 = 40 − 16 = 24

Calculating Acceleration at t=5s Next, we find the instantaneous acceleration at t = 5 s by substituting t = 5 into the acceleration function.

a ( 5 ) = 8

Finding Time When Stationary To find the time when the particle is stationary, we need to set the velocity function equal to zero and solve for t .

v ( t ) = 8 t − 16 = 0
8 t = 16
t = 8 16 ​ = 2

Final Answer Therefore, the instantaneous velocity at t = 5 s is 24, the instantaneous acceleration at t = 5 s is 8, and the particle is stationary at t = 2 s.

Examples
Understanding the motion of objects is crucial in many fields. For example, in sports, analyzing the position, velocity, and acceleration of a ball or an athlete can help improve performance. Similarly, in engineering, understanding the motion of mechanical parts is essential for designing efficient and safe machines. This problem demonstrates how calculus can be used to analyze motion and make predictions about the behavior of moving objects.

Answered by GinnyAnswer | 2025-07-04