Basic kinematics variables using calculus: 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

First, we need to find the equation for position as a function of time using the given velocity equation. To do this, we can integrate the velocity equation:

v(t) = dx/dt 
dx = v(t) dt 
integrate both sides: 
x(t) = ∫(3.00 m/s + (4.00 m/s^2)t) dt 
x(t) = 3.00 m/s*t + (4.00 m/s^2)(t^2)/2 + C 
where C is the constant of integration.

We can find the value of C by using the initial condition x(0) = 1.00 m:
x(0) = 3.00 m/s*0 + (4.00 m/s^2)(0^2)/2 + C 
C = 1.00 m

Therefore, the equation for position as a function of time is:
x(t) = 3.00 m/s*t + (4.00 m/s^2)(t^2)/2 + 1.00 m

Simplifying this equation, we get:
x(t) = 1.00 m + (3.00 m/s)t + (1.33 m/s^3)t^3

So the answer is choice E.

Basic Kinematics Variables Using Calculus: the Velocity of an Object