The velocity of an object is given by the expression v(t) = 3.00 m/s + (4.00 m/s³)t², where t is in seconds. Determine the position of the object as a function of time if it is located at x = 1.00 m at time T = 0.000 s. A) (4.00 m/s)t + 1.00 m B) (3.00 m/s)t + (1.33 m/s3)t³ C) (4.00 m/s)t D) 1.33 m E) 1.00 m + (3.00 m/s)t + (1.33 m/s³)t³

Question

Quizplus · Accepted Answer

To find the position of the object as a function of time, we need to integrate the velocity function with respect to time. 
∫v(t) dt = ∫(3.00 m/s + (4.00 m/s^2)t) dt
= 3.00t + (2.00 m/s^2)t^2 + C
where C is the constant of integration. To find C, we use the initial condition that the object is located at x = 1.00 m at time t = 0.00 s: 
1.00 m = 3.00(0.00) + (2.00 m/s^2)(0.00)^2 + C
C = 1.00 m
So, the position function is: 
x(t) = 3.00t + (2.00 m/s^2)t^2 + 1.00 m
Rearranging and simplifying, we get: 
x(t) = 1.00 m + (3.00 m/s)t + (2.00 m/s^2)t^2
Using the identity 2a + b = (a+b)^2 - (a-b)^2 with a = 1.33 m/s^2 and b = 3.00 m/s, we can write: 
x(t) = 1.00 m + (3.00 m/s)t + (1.33 m/s^2)t^2 + (3.00 m/s)(1.33 m/s^2)t^3
= 1.00 m + (3.00 m/s)t + (1.33 m/s^3)t^3
Therefore, the best choice is E.

The Velocity of an Object Is Given by the Expression