speed and velocity question

Question

An baseball travels horizontally with a velocity v below. How far does it roll between t = 0 and t = 4 seconds?

 

v(t)=2t+5

Answer 1

The velocity is the instantaneous time rate of change of the position. Let x(t) be the position, this means that v(t) = dx/dt. That is, v(t) is the first time derivative of the position. To get the position from the velocity, one must integrate the velocity equation. We want to know how far the ball traveled from t = 0 to t = 4, so we integrate v(t)dt from t = 0 to t = 4 sec. The integral of 2t + 5 is t^2 + 5t. Evaluating the integral from t = 0 to t = 4 we get 4^2 + (5)(4) - 0^2 - (5)(0) = 16 + 20 = 36.

 

Velocity units were not specified in this problem, so I don't know if this is 36 meters or 36 feet or whatever...