v(t) = 2t + 1 m/s from t = 0 to t = 3. Find the displacement and the… Step by Step

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The question

Question: find the displacement and the total distance for v(t) = 2t + 1 from t = 0 to t = 3.

v(t) = 2t + 1

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 3 of (2t + 1) dt

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

Now we have t² + t.

= (t² + t) from 0 to 3

Put in 3, then take away the same with 0.

= (3² + 3) − (0² + 0)

That is 12 minus 0: 12.

= 12 − 0 = 12

So the displacement is 12 m.

displacement = 12 m

2 Where it turns around

v never changes sign between 0 and 3.

It never turns around.

v does not change sign between 0 and 3

Draw v.

From 0 to 3, v is positive: it moves forward.

3 The total distance

From 0 to 3: the integral is 12, the same way as before.

∫ from 0 to 3 of (2t + 1) dt = 12

Distance counts every piece as positive.

Add them: 12 m.

distance = |12| = 12 m

4 Check

Let's check: one piece, all one way.

The distance and the displacement agree.

Right!

12 = 12

Let's look back.

First: the net change.

Then: where it turns around.

Then: the total distance.

Last: check.

Answer