The acceleration of an object (in m/s2) is given by the function a(t) = 3 sin ( t ) .

Question

The initial velocity of the object v(0)=-5 m/s. Round your answers to four decimal places.

a) Find an equation v(t) for the object velocity.

b) Find the object's displacement (in meters) from time 0 to time 3.

c) Find the total distance traveled by the object from time 0 to time 3.

Answer 1

a(t) = 3sint

integrate to get velocity, where c is an unknown constant

v(t) = -3cost + c

plug in t=0 to solve for c

v(0) = -3cos(0) + c = 5

c = 5+3(1) = 8

v(t) = -3cost + 8

integrate again to find displacement

d(t) = 3sint + 8t + k where k= an unknown constant

evaluted from 0 to 3

d(3) = 3sin(3) +8(3)

for distance, use the arc length formula

s= integral of sqr(1-dy/dx) evaluated from 0 to 3

with y=3sint, y'=dy/dx=-3costt