Question

The acceleration of an object (in m/s ^2 ) is given by the function a(t)=6sin(t). The initial velocity of the object is v(0)=−11 m/s. Round your answers to four decimal places. a) Find an equation v(t) for the object velocity. v(t)= b) Find the object's displacement (in meters) from time 0 to time 3. meters c) Find the total distance traveled by the object from time 0 to time 3 . meters

Answer 1

a) The equation for the object's velocity is v(t)=−6cos(t)+11 m/s.

b) The object's displacement from time 0 to time 3 is approximately 14.7921 meters.

c) The total distance traveled by the object from time 0 to time 3 is approximately 27.0637 meters.

Step-by-step explanation:

a) To find the velocity equation v(t), we need to integrate the given acceleration function a(t)=6sin(t) with respect to time t. The antiderivative of sin(t) is −cos(t). Therefore, v(t)=−6cos(t)+C, where C is the constant of integration. To determine C, we use the initial velocity v(0)=−11 m/s. Substituting this into the equation, we get C=11, and the final velocity equation is v(t)=−6cos(t)+11 m/s.

b) To find the displacement, we integrate the velocity function v(t) with respect to time from 0 to 3: ∫³₀ (−6cos(t)+11)dt. Evaluating this definite integral gives us the displacement, which is approximately 14.7921 meters.

c) To find the total distance traveled, we consider both the positive and negative parts of the velocity function. We take the integral of the absolute value of v(t) from 0 to 3: ∫³₀∣(−6cos(t)+11)∣dt. Evaluating this integral yields the total distance traveled, which is approximately 27.0637 meters.

Answer 2

a) The equation for the object's velocity is v(t) = -6cos(t) - 5.
b) The object's displacement from time 0 to time 3 is approximately -15.8466 meters.
c) The total distance traveled by the object from time 0 to time 3 is approximately 15.8466 meters.

a) To find the velocity function v(t), integrate the acceleration function a(t) with respect to time t, and then add the constant of integration based on the initial velocity.

v(t) = ∫ a(t) dt = ∫ 6sin(t) dt = -6cos(t) + C

To find C, use the initial velocity v(0) = -11 m/s:
v(0) = -6cos(0) + C = -6 + C = -11

Solving for C, we get C = -5. Therefore, the equation for the object's velocity is:
v(t) = -6cos(t) - 5

b) To find the object's displacement from time 0 to time 3, integrate the velocity function over that time interval:
Displacement = ∫03 v(t) dt = ∫03 (-6cos(t) - 5) dt

Using calculus or a calculator, evaluate this definite integral to get the displacement.
Displacement ≈ -15.8466 meters

c) To find the total distance traveled by the object from time 0 to time 3, take the integral of the absolute value of the velocity function over that time interval:
Total Distance = ∫03 |v(t)| dt = ∫03 |-6cos(t) - 5| dt

Again, use calculus or a calculator to evaluate this definite integral to get the total distance.
Total Distance ≈ 15.8466 meters