Change in Position Examples

A particle moving along the x-axis is at position x = 6 when t = 0. If

where is the particle when t = 5?

The problem gives us s(0) = 6 and the value of the integral and asks for s(5). We know

so we plug in the value of the integral and the value of s(0) to get

10 = s(5) – 6.

Solving, we get

s(5) = 10 + 6 = 16.

We know the particle starts at x = 6 and moves a distance of 10 from time 0 to time 5. This means the particle must end up at position x = 16 at time t = 5.

A cat climbing up and down a tree ends at height 3 feet after 10 seconds. If  feet, how high in the tree was the cat at time t = 0?

The problem gives us s(10) = 3 feet and the value of the integral and asks for s(0).

We know

so we plug in the value of the integral and the value of s(10) to get

-16 = 3 – s(0).

Solving, we get

s(0) = 3 + 16 = 19.

This means the cat started out 19 feet up in the tree.

In pictures, we know the cat ended up at height 3 when t = 10, and we know the cat climbed down 16 feet to get there because

This means the cat must have started 19 feet high when t = 0.

Be Careful when you're given s(b) and the value of . The integral tells you how much you had to change from the starting position s(a) in order to get to s(b).

To get back to s(a) you have to reverse the direction.

Let s(b) = 10.

(a) If , that means we had to take 3 away from s(a) in order to get to s(b).

(b) If , that means we had to add 7 to s(a) in order to get to s(b).

(a)To get back to s(a), we have to add 3 to s(b), so

s(a) = 10 + 3 = 13.

(b) To get back to s(a), we have to subtract 7 from

s(b), so

s(a) = 10 – 7 = 3.

You could also be given the values of s(a) and s(b) and asked for the value of the integral.

A bicyclist is 10 miles from home after 3 hours of riding and 30 miles from home after 5 hours of riding. If her velocity is given by the function v(t), what is ?

The problem tells us s(3) = 10 and s(5) = 30, and asks us for . Since

and

change in position between time 3 and time 5 = s(5) – s(3),

we have

Be Careful: When a problem has units, remember to put units in your answer. Units make your teacher happy, and that makes you happy!

If s(0) = 3 and , what is s(6)?

Using the rearranged equation,

We plug in the given values to get

3 + 4 = s(6),

so s(6) = 7.