The acceleration of an object (in m/s2) is given by the function a ( t ) = 4 sin ( t ) . The initial velocity of the object is v ( 0 ) = − 3 m/s. Round your answers to four decimal places.

Question

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

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) v(t) = ∫4sintdt = - 4cost +C; v(0) = - 4cos0 + C = -3; C = - 3 + 4 = 1; v(t) = - 4cost + 1

b) displacement s = ∫₀³(- 4cost + 1)dt = (- 4sint + t)₀³ = - 4sin3 + 3 = 2.4355.

c)Distance d = ∫₀³I- 4sint + tIdt = 4.7019 (for this integral we used TI-84)

Answer 2

a) v = integral of adt. solve for the integration constant by setting v(0)=-3

b) d = integral of vdt with no constant. the constant would be necessary if solving for position.

c) In order to find the distance travelled, you need to find the zeroes of v(t). Find d for the intervals for positive v and add -d for the intervals of negative v. Distance is how far is travelled of the integral of speed * dt which is |v|dt.