Question

1)a)The acceleration of a car is given by the function a(t) = sin(t) m / s² at time t s. The average acceleration for 0 ≤ t ≤ π s is _____ m / s². Round your answer to two decimal places.

b) The acceleration is given by a(t) = 4t at time t s. The initial position is 1 m, and, the initial velocity is 3 m / s. At time t = 4 s, the position is _____ m and the velocity is _____ m / s. Give both answers to two decimal places.

Answer 1

1a)

Average of acceleration = $\int_{o}^{\prod }$ sin(t)dt /{π-0}

Average=1/π { -(Cost)^π0 } = - ( -1-(+1)) /π= 2/π

Average acceleration from 0 to π second = 2/π m/s ²

= 0.64 m/s ²

a(t) = 4t ; X_{​​​​​​0} = 1 m ; u​​​​​​_{​​​​​​o} = 3 m/s

Integrating the acceleration, we get

V(t) = 2 t ² + C ; where c is integration constant

Put t = 0 for using intial values

3 = 0 + c

Therefore value of c = 3 m/s

Therefore velocity at t= 4 seconds is given by

V(t=4seconds) = 2(4)² + 3 = 35 m/s

Now , integrating the v(t) , we get the position function as :

X(t) = 2t³/3 + 3t + D ; where D is constant of integration

Put t = 0 for using initial values

1 = 0 + D

Therefore value of D = 1 m

Therefore the position at t = 4 seconds

x( t= 4 seconds) = 2(4)³/3 + 3×4 + 1

= 42.66 + 13 = 55.67 m