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Linear Motion

Physics · Linear Motion

Acceleration

The video lesson for this topic is coming soon.

Essential Question

How do we describe a change in velocity?

A car pulling away from a stop sign and a bike braking at the bottom of a hill both have a changing velocity. By the end of this lesson, you should be able to calculate acceleration, use it to find a final velocity, and tell whether an object is speeding up or slowing down.

Acceleration

  • How fast velocity changes
  • Speeding up or slowing down
  • Changing direction counts too
Three snapshots of a car, one second apart. At 0 s its velocity arrow is labeled 2 m/s, at 1 s it is longer and labeled 4 m/s, and at 2 s it is longest and labeled 6 m/s. Caption: velocity grows by 2 m/s every second.

Acceleration describes how quickly an object’s velocity changes. This car’s velocity goes from 2 m/s to 4 m/s to 6 m/s, gaining 2 m/s every second, so it is accelerating. Slowing down is also acceleration, and so is changing direction, because velocity includes direction.

The Acceleration Equation

a = (vf − vi) / t

  • a = acceleration, in meters per second squared (m/s2)
  • vf = final velocity, in m/s
  • vi = initial velocity, in m/s
  • t = time, in seconds (s)
The equation a = (v_f − v_i) / t in a yellow box. Below it, four cards: a is acceleration in m/s squared, v_f is final velocity in m/s, v_i is initial velocity in m/s, and t is time in s.

Acceleration is the change in velocity, final minus initial, divided by the time it took. Always subtract in the order final minus initial, so the sign of your answer comes out right.

Units of Acceleration

  • (m/s) ÷ s = m/s2
  • Meters per second, every second
  • 2 m/s2: gain 2 m/s each second
A table of time and velocity. At 0 s, 0 m/s. At 1 s, 2 m/s. At 2 s, 4 m/s. At 3 s, 6 m/s. Arrows between the rows are labeled +2 m/s. Caption: a = 2 m/s squared.

Dividing a velocity (m/s) by a time (s) gives meters per second per second, written m/s2. It’s easiest to read as “meters per second, every second.” An acceleration of 2 m/s2 means the velocity changes by 2 m/s each second, as the table shows.

Speeding Up vs. Slowing Down

  • a same direction as v: speeding up
  • a opposite to v: slowing down
Two cars moving right. In the top car, the blue velocity arrow and the green acceleration arrow both point right, labeled speeding up. In the bottom car, the velocity arrow points right but the acceleration arrow points left, labeled slowing down.

Whether an object speeds up or slows down depends on how its acceleration compares to its velocity. If the acceleration points the same way as the velocity, the object speeds up. If the acceleration points the opposite way, the object slows down, like a car braking while it still rolls forward.

The Sign of Acceleration

  • Sign = direction of a
  • Same signs: speeding up
  • Opposite signs: slowing down
A two by two grid. Columns are a positive and a negative. Rows are v positive and v negative. Positive v with positive a: speeding up. Positive v with negative a: slowing down. Negative v with positive a: slowing down. Negative v with negative a: speeding up.

In one-dimensional problems, the sign of the acceleration tells you its direction, not whether the object is speeding up. Compare the signs of v and a. If they match, the object speeds up. If they are opposite, it slows down. So a negative acceleration can mean speeding up, for example when an object moving in the negative direction goes faster and faster.

Finding Final Velocity

vf = vi + at

  • Start: a = (vf − vi)/t
  • Multiply both sides by t
  • Add vi to both sides
The equation v_f = v_i + at in a yellow box. Below it, labels explain that v_f is where you end, v_i is where you start, and at is how much the velocity changes.

Rearranging the acceleration equation gives vf = vi + at. Multiply both sides by t to get at = vf − vi, then add vi to both sides. The product at is how much the velocity changes, so the final velocity is the starting velocity plus that change.

Worked Examples

Try each problem on your own first, then check your work against the solution.

Example 1: Speeding Up

A car speeds up from 5.0 m/s east to 25.0 m/s east in 4.0 s. What is its acceleration?

A light blue car labeled 5.0 m/s on the left and a dark blue car labeled 25.0 m/s on the right, with a double arrow between them labeled 4.0 s and an arrow labeled east.
  1. Knowns: vi = 5.0 m/s east, vf = 25.0 m/s east, t = 4.0 s
  2. Equation: a = (vf − vi)/t
  3. Substitute: a = (25.0 m/s − 5.0 m/s) / 4.0 s = 20.0 m/s / 4.0 s
  4. Answer: a = 5.0 m/s2 east

Example 2: Slowing Down

A cyclist moving 12 m/s east brakes to a stop in 3.0 s. What is her acceleration?

A cyclist moving right toward a red stop sign, with a blue arrow labeled 12 m/s east.
  1. Knowns: vi = 12 m/s east, vf = 0 m/s (a stop), t = 3.0 s
  2. Equation: a = (vf − vi)/t
  3. Substitute: a = (0 m/s − 12 m/s) / 3.0 s = −12 m/s / 3.0 s
  4. Answer: a = −4.0 m/s2, which is 4.0 m/s2 west

The acceleration is negative while the velocity is positive. Opposite signs mean she slows down, which is exactly what braking does.

Example 3: Finding Final Velocity

A skateboarder rolling at 2.0 m/s speeds up at 1.5 m/s2 down a ramp for 4.0 s. What is her final velocity?

A stick figure skateboarder at the top of a ramp that slopes down to the right, with a green arrow along the ramp labeled a = 1.5 m/s squared.
  1. Knowns: vi = 2.0 m/s, a = 1.5 m/s2, t = 4.0 s
  2. Equation: vf = vi + at
  3. Substitute: vf = 2.0 m/s + (1.5 m/s2)(4.0 s) = 2.0 m/s + 6.0 m/s
  4. Answer: vf = 8.0 m/s down the ramp

Example 4: Finding Time

A train speeds up from 10.0 m/s to 30.0 m/s with an acceleration of 0.50 m/s2. How long does this take?

A two-car train on a track, with labels 10.0 m/s to 30.0 m/s and a = 0.50 m/s squared.
  1. Knowns: vi = 10.0 m/s, vf = 30.0 m/s, a = 0.50 m/s2
  2. Equation: a = (vf − vi)/t, so t = (vf − vi)/a
  3. Substitute: t = (30.0 m/s − 10.0 m/s) / 0.50 m/s2 = 20.0 m/s / 0.50 m/s2
  4. Answer: t = 40. s

The decimal point in “40.” shows that both digits are significant, matching the two significant figures in 0.50 m/s2.

Big Ideas

  • a = (vf − vi)/t, measured in m/s2
  • vf = vi + at
  • a same direction as v: speeding up
  • a opposite to v: slowing down

Back to the essential question: we describe a change in velocity with acceleration, the change in velocity divided by the time it takes, a = (vf − vi)/t, measured in m/s2. When acceleration points the same way as velocity, the object speeds up. When it points the opposite way, the object slows down. Next up, we’ll see what acceleration looks like on a velocity-time graph.

Take It With You

Printable lesson slides and a practice worksheet for this topic are coming soon.