v(t) = t^2 - 4t + 3 m/s from t = 0 to t = 4. Find the displacement… Step by Step

Step by step

The question

Question: find the displacement and the total distance for v(t) = t² − 4t + 3 from t = 0 to t = 4.

v(t) = t² − 4t + 3

Displacement is where it ends up, compared with the start.

Distance counts every step, forward and back.

displacement: where it ends up

distance: how far it travels

1 The net change

The displacement is the integral of v.

Going back counts as negative.

displacement = ∫ from 0 to 4 of (t² − 4t + 3) dt

The antiderivative: add 1 to each power, divide by the new power.

Now we have t³ over 3 − 2 t² + 3t.

= (t³3 − 2t² + 3t) from 0 to 4

Put in 4, then take away the same with 0.

= (4³3 − 2(4)² + 3(4)) − (0³3 − 2(0)² + 3(0))

That is 4/3 minus 0: 4/3.

= 43 − 0 = 43

So the displacement is 4/3 m.

displacement = 43 m

2 Where it turns around

It turns around where v changes sign.

First, where is v = 0?

t² − 4t + 3 = 0

We need two numbers that multiply to 3 and add to −4: −1 and −3.

(t − 1)(t − 3) = 0

v is 0 at t = 1 and t = 3.

t = 1 or 3

Draw v.

From 0 to 1, v is positive: it moves forward.

From 1 to 3, v is negative: it moves back.

From 3 to 4, v is positive: it moves forward.

3 The total distance

From 0 to 1: the integral is 4/3, the same way as before.

∫ from 0 to 1 of (t² − 4t + 3) dt = 43

From 1 to 3: the integral is −4/3, the same way as before.

∫ from 1 to 3 of (t² − 4t + 3) dt = −43

From 3 to 4: the integral is 4/3, the same way as before.

∫ from 3 to 4 of (t² − 4t + 3) dt = 43

Careful!

The integral of v is the displacement.

Going back cancels going forward.

For the distance, add each piece without its sign.

Careful: not distance = 43 ✗

Distance counts every piece as positive.

Add them: 4 m.

distance = |43| + |−43| + |43| = 4 m

4 Check

Let's check: the pieces with their signs add up to 4/3, the displacement.

Right!

43 − 43 + 43 = 43

Let's look back.

First: the net change.

Then: where it turns around.

Then: the total distance.

Last: check.

Answer