Can anyone help with problem 2 on HW 15? The question is: The velocity function (in meters per second) for a particle moving along a line is given by 𝑣(𝑡)=𝑡^3−4𝑡^2 Find the displacement and the distance traveled by the particle during the time interval [-2,6].
I did make a graph of the velocity function like the question suggests but I’m still lost. Any help would be great, thanks.
Well, this is a hard one! Let’s take a look at a graph of the velocity function over the time interval [-2,6]:
Here’s the thing to notice: The velocity is negative for t<4 and positive for t>4. Thus the object is moving to the left until t=4 and then back to the right after that. You can find out how far it travels over each of those portions using the anti-derivative technique. That is, if we take
then the total displacement up until t=4 should be p(4)-p(-2); this should be a negative number. The total displacement after t=4 should be a positive number. The sum of those two numbers (which might be positive or negative) would be the total displacement. The sum of the absolute values of those numbers (which is definitely positive) would be the total distance traveled.
