The equation of motion is given for a particle, where [tex]$s$[/tex] is in meters and [tex]$t$[/tex] is in seconds. Find the acceleration after 5.5 seconds.
[tex]$s=\sin 2 \pi t$[/tex]
Options:
[tex]$166.375 \pi^2 M / s ^2$[/tex]
[tex]$0 M / s ^2$[/tex]
[tex]$30.25 \pi M / S ^2$[/tex]
[tex]$-30.25 \pi M / s ^2$[/tex]
[tex]$-166.375 \pi^2 M / s ^2$[/tex]
Answer (2)
Find the velocity function by taking the first derivative of the position function: v ( t ) = 2 π cos ( 2 π t ) .
Find the acceleration function by taking the derivative of the velocity function: a ( t ) = − 4 π 2 sin ( 2 π t ) .
Evaluate the acceleration function at t = 5.5 seconds: a ( 5.5 ) = − 4 π 2 sin ( 11 π ) .
Since sin ( 11 π ) = 0 , the acceleration at t = 5.5 seconds is 0 .
Explanation
Problem Analysis We are given the equation of motion for a particle as s = sin ( 2 π t ) , where s is in meters and t is in seconds. Our goal is to find the acceleration of the particle at t = 5.5 seconds.
Finding the Velocity Function To find the acceleration, we need to find the second derivative of the position function s ( t ) with respect to time t . The first derivative will give us the velocity function, and the second derivative will give us the acceleration function.
Calculating the Velocity First, let's find the velocity function v ( t ) by taking the derivative of s ( t ) with respect to t :
v ( t ) = d t d s = d t d ( sin ( 2 π t )) Using the chain rule, we get: v ( t ) = 2 π cos ( 2 π t )
Calculating the Acceleration Now, let's find the acceleration function a ( t ) by taking the derivative of v ( t ) with respect to t :
a ( t ) = d t d v = d t d ( 2 π cos ( 2 π t )) Again, using the chain rule, we get: a ( t ) = − 4 π 2 sin ( 2 π t )
Evaluating at t=5.5 Now we need to evaluate the acceleration function at t = 5.5 seconds: a ( 5.5 ) = − 4 π 2 sin ( 2 π ( 5.5 )) = − 4 π 2 sin ( 11 π ) Since sin ( 11 π ) = 0 , we have: a ( 5.5 ) = − 4 π 2 ( 0 ) = 0
Final Answer Therefore, the acceleration of the particle at t = 5.5 seconds is 0 m/s 2 .
Examples
Understanding the motion of objects is crucial in many real-world applications. For example, when designing a suspension system for a car, engineers need to calculate the acceleration of the car at different points in time to ensure a smooth and comfortable ride. Similarly, in robotics, understanding the acceleration of a robot's joints is essential for precise movements and avoiding collisions. This problem demonstrates how calculus can be used to analyze motion and make predictions about the behavior of objects.
The acceleration of the particle at t = 5.5 seconds is 0 m/s², as calculated from the position function s = sin(2πt). The sine function evaluates to zero at this time, leading to a zero acceleration. Thus, the correct answer is 0 M / s².
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