A particle is moving along a straight line and the acceleration is given by a(t)=3t+5, (units m/s2). a)Find v(t) , the velocity at time t if the initial velocity is v(0) = -4 ft/s? b)Find the total distance traveled during the time interval [0, 2]?
Note: I apologize in advance for the long solution; I am not sure how much one who reads this understands and have attempted to give some explanation for each step taken. Hope this helps!
Let's begin by writing out what we are given:
Before we begin to solve the problem, we need to convert acceleration and velocity into some common unit. In this solution, we will be using SI (
Integrating both sides, we have:
Applying the above formula, we can find
Here, we must remember how to integrate a polynomial:
Since we are given that
To answer part b), we use the
However, we can take a shortcut and notice that in the quadratic, both the
Here, we need to remember that
Since we have that
Using the Fundamental Theorem of Calculus, we have that:
Plugging this into a calculator we have: