Distance and displacement from velocity

About this lesson

Calculus BC (AP-aligned) › Integrals and accumulation

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Distance and displacement from velocity

Warm-up

Three quick questions from earlier lessons.

Discover

A cyclist rides at a steady 12 mph for 2 hours. On a graph of velocity against time, that is a horizontal line at 12 from t=0t = 0 to t=2t = 2. What is the area of the rectangle under it, and what does it mean?

  1. 24: the distance in miles
  2. 14: the sum of the numbers

Key idea

Area under a rate graph gives the total change.

If v(t)≥0v(t) \ge 0 is a velocity, the area between its graph and the tt-axis on [a,b][a, b] is the distance traveled. If vv is sometimes negative, areas below the axis count negatively and the signed total is the displacement.

Worked example

v(t)=3tv(t) = 3t m/s. How far does the object go from t=0t = 0 to t=4t = 4?

  1. Step 1The velocity grows steadily: v(t) = 3t.
  2. Step 2The distance is the area under the graph from 0 to 4.
  3. Step 3A triangle: ½ · 4 · 12 = 24 meters.

Base 44, height v(4)=12v(4) = 12
The region under v is a triangle.
Area =12⋅4⋅12=24= \tfrac{1}{2} \cdot 4 \cdot 12 = 24
½ × base × height.
Distance: 2424 m
Units: m/s × s.

Your turn, guided

Your turn: 30 mph for 2 hours, then 50 mph for 1 hour.

Distance: 110110 miles
60 + 50.

Practice ★

  1. A car drives at 30 mph for 2 hours, then at 57 mph until hour 5. How far does it travel, in miles?
  2. A car drives at 28 mph for 3 hours, then at 34 mph until hour 4. How far does it travel, in miles?
  1. From rest, a cart's velocity is v(t)=4tv(t) = 4t m/s. How far does it go in the first 6 seconds, in meters?
  2. From rest, a cart's velocity is v(t)=9tv(t) = 9t m/s. How far does it go in the first 5 seconds, in meters?

Watch out!

Practice ★★

  1. A particle's velocity is v(t)=t−1v(t) = t - 1 m/s. Find its displacement from t=0t = 0 to t=8t = 8, in meters (areas below the axis count as negative).
  2. A particle's velocity is v(t)=t−3v(t) = t - 3 m/s. Find its displacement from t=0t = 0 to t=5t = 5, in meters (areas below the axis count as negative).

Exit ticket

Three questions to check you have got it.

  1. From rest, a cart's velocity is v(t)=4tv(t) = 4t m/s. How far does it go in the first 6 seconds, in meters?
  2. From rest, a cart's velocity is v(t)=9tv(t) = 9t m/s. How far does it go in the first 5 seconds, in meters?
  3. From rest, a cart's velocity is v(t)=7tv(t) = 7t m/s. How far does it go in the first 3 seconds, in meters?

Summary card

Distance from velocity

  • Area under vv = distance (when v≥0v \ge 0).
  • Signed area (below the axis negative) = displacement.

Watch out: use the right area formula, ½ for triangles.

End of the lesson

Next lesson: Riemann sums Back to the chapter

Answers: Distance and displacement from velocity

A

  1. 231231
  2. 118118

B

  1. 7272
  2. 2252\frac{225}{2}

C

  1. 2424
  2. −52- \frac{5}{2}

D

  1. 7272
  2. 2252\frac{225}{2}
  3. 632\frac{63}{2}

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