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.
1a)
Average of acceleration = 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 ; X0 = 1 m ; uo = 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
1)a)The acceleration of a car is given by the function a(t) = sin(t) m / s²...
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