The acceleration of an object (in [tex]$m / s ^2$[/tex]) is given by the function [tex]$a(t)=3 inos(t)$[/tex]. The initial velocity of the object is [tex]$v(0)=-2 m / s$[/tex]. Round your answers to four decimal places.
a) Find an equation [tex]$v(t)$[/tex] for the object velocity.
[tex]$v(t)= \square$[/tex]
b) Find the object's displacement (in meters) from time 0 to time 3.
[tex]$\square$[/tex] meters
c) Find the total distance traveled by the object from time 0 to time 3.
[tex]$\square$[/tex] meters
Answer (1)
The velocity function is found by integrating the acceleration function: v ( t ) = ∫ a ( t ) d t = − 3 cos ( t ) + 1 .
Displacement from t = 0 to t = 3 is calculated by integrating the velocity function over the interval [ 0 , 3 ] : ∫ 0 3 v ( t ) d t = − 3 sin ( 3 ) + 3 ≈ 2.5766 .
Total distance traveled requires finding when v ( t ) = 0 , which occurs at t 1 = arccos ( 3 1 ) . The total distance is then calculated as ∫ 0 t 1 ∣ v ( t ) ∣ d t + ∫ t 1 3 ∣ v ( t ) ∣ d t = 4 2 − 2 arccos ( 3 1 ) − 3 sin ( 3 ) + 3 ≈ 4.3253 .
The velocity equation is v ( t ) = − 3 cos ( t ) + 1 , displacement is 2.5766 meters, and total distance traveled is 4.3253 meters. v ( t ) = − 3 cos ( t ) + 1 , Displacement = 2.5766 m , Distance = 4.3253 m
Explanation
Problem Setup We are given the acceleration function a ( t ) = 3 sin ( t ) and the initial velocity v ( 0 ) = − 2 m/s. We need to find the velocity function v ( t ) , the displacement from time 0 to time 3, and the total distance traveled from time 0 to time 3.
Finding Velocity Function To find the velocity function v ( t ) , we integrate the acceleration function a ( t ) with respect to t :
v ( t ) = ∫ a ( t ) d t = ∫ 3 sin ( t ) d t = − 3 cos ( t ) + C
Applying Initial Condition We use the initial condition v ( 0 ) = − 2 to find the constant C :
− 2 = − 3 cos ( 0 ) + C = − 3 ( 1 ) + C
C = 1
Therefore, the velocity function is v ( t ) = − 3 cos ( t ) + 1 .
Calculating Displacement To find the displacement from time 0 to time 3, we integrate the velocity function v ( t ) from 0 to 3:
Displacement = ∫ 0 3 v ( t ) d t = ∫ 0 3 ( − 3 cos ( t ) + 1 ) d t = [ − 3 sin ( t ) + t ] 0 3 = ( − 3 sin ( 3 ) + 3 ) − ( − 3 sin ( 0 ) + 0 ) = − 3 sin ( 3 ) + 3
Using a calculator, we find that − 3 sin ( 3 ) + 3 ≈ 2.5766 .
Finding Times When Velocity is Zero To find the total distance traveled, we need to find the times when the velocity is zero to determine when the object changes direction.
v ( t ) = − 3 cos ( t ) + 1 = 0
cos ( t ) = 3 1
Let t 1 = arccos ( 3 1 ) . Since 0 ≤ t ≤ 3 , we need to check if t 1 is in the interval [ 0 , 3 ] . Since arccos ( 3 1 ) ≈ 1.231 , t 1 is in the interval.
Setting up Integral for Total Distance Since v ( 0 ) = − 2 < 0 , v ( t ) is negative near t = 0 . Thus, v ( t ) < 0 on [ 0 , t 1 ] and 0"> v ( t ) > 0 on [ t 1 , 3 ] . So, the total distance is
∫ 0 3 ∣ v ( t ) ∣ d t = ∫ 0 t 1 − v ( t ) d t + ∫ t 1 3 v ( t ) d t = ∫ 0 t 1 ( 3 cos ( t ) − 1 ) d t + ∫ t 1 3 ( − 3 cos ( t ) + 1 ) d t
Calculating Total Distance Calculate the integrals:
∫ 0 t 1 ( 3 cos ( t ) − 1 ) d t = [ 3 sin ( t ) − t ] 0 t 1 = 3 sin ( t 1 ) − t 1
∫ t 1 3 ( − 3 cos ( t ) + 1 ) d t = [ − 3 sin ( t ) + t ] t 1 3 = ( − 3 sin ( 3 ) + 3 ) − ( − 3 sin ( t 1 ) + t 1 ) = − 3 sin ( 3 ) + 3 + 3 sin ( t 1 ) − t 1
Total distance = 3 sin ( t 1 ) − t 1 − 3 sin ( 3 ) + 3 + 3 sin ( t 1 ) − t 1 = 6 sin ( t 1 ) − 2 t 1 − 3 sin ( 3 ) + 3 . Since cos ( t 1 ) = 3 1 , sin ( t 1 ) = 1 − cos 2 ( t 1 ) = 1 − 9 1 = 9 8 = 3 2 2 . Therefore, the total distance is
6 ( 3 2 2 ) − 2 arccos ( 3 1 ) − 3 sin ( 3 ) + 3 = 4 2 − 2 arccos ( 3 1 ) − 3 sin ( 3 ) + 3 ≈ 4.3253
Final Answer a) The equation for the object's velocity is v ( t ) = − 3 cos ( t ) + 1 .
b) The object's displacement from time 0 to time 3 is approximately 2.5766 meters.
c) The total distance traveled by the object from time 0 to time 3 is approximately 4.3253 meters.
Examples
Understanding motion, as described by acceleration, velocity, and displacement, is crucial in many real-world applications. For example, when designing a suspension system for a car, engineers need to consider the forces acting on the car (acceleration), how fast the car is moving (velocity), and how far the car travels (displacement) to ensure a smooth and safe ride. Similarly, in robotics, controlling the motion of a robot arm requires precise calculations of acceleration, velocity, and displacement to perform tasks accurately. These concepts are also fundamental in fields like aerospace engineering, where understanding the motion of aircraft and spacecraft is essential for navigation and control.
