Final answer:
The initial velocity, given that displacements at t=0 and t=2 s are equal and the acceleration function is a=1-2t, can be found using kinematic equations and is 0.66 m/s.
Thus option B. is correct answer.
Step-by-step explanation:
To find the initial velocity of an object with a time-varying acceleration, we can use kinematic equations. Given that the acceleration a = 1 - 2t, we can integrate this expression to find the velocity function, and then, by using the condition that displacements at t = 0 and t = 2 seconds are equal, solve for the initial velocity. To integrate the acceleration, we get:
- v(t) = ∫ a dt = ∫ (1 - 2t) dt = t - t² + C
Where C is the constant of integration, which in this case, is the initial velocity v0. Then we calculate the displacement by integrating the velocity:
- x(t) = ∫ v dt = ∫ (t - t² + C) dt = ½ t² - ⅓ t³ + Ct + D
Where D is the constant of integration representing the initial displacement, which we can set as zero since it does not affect the comparison of displacements at different times. We now set x(2) = x(0), which gives:
- x(2) = ½(4) - ⅓(8) + 2C = x(0) = 0
Solving for C gives us the initial velocity:
- 2C = ½(4) - ⅓(8)
- C = -½(4) + ⅓(4)
- C = -2 + ⅓(4) = -2 + ⅓(4) = -2 + 2.67 = 0.67 m/s
The correct answer is B. 0.66 m/s.