Speed, Velocity, and Acceleration

7.1 Measuring How Fast Objects Move

The most natural question to ask about a moving object is: how fast is it going? Informally, fast means covering a lot of distance in a short time. Formally, speed is the ratio of distance traveled to the time taken. This ratio — a distance divided by a time — gives a single number that tells us how rapidly position is changing, regardless of direction.

Speed has units of distance per time: meters per second (m/s) in SI, kilometers per hour (km/h) in everyday life, or miles per hour (mph) in some countries. Converting between these is a dimensional analysis problem (Chapter 3). In physics, m/s is the standard unit.

7.2 Average Speed

Average speed is the total distance traveled divided by the total time elapsed. It is a scalar quantity: it has magnitude but no direction. Average speed is always positive, or zero if the object does not move.

average speed = total distance / total time = d / Δt

7.3 Instantaneous Speed

Average speed tells you how fast an object moved over an interval of time. Instantaneous speed tells you how fast it is moving at one specific moment. A car’s speedometer displays instantaneous speed — not average speed.

Formally, instantaneous speed is the limit of average speed as the time interval shrinks to zero. As the time interval Δt approaches zero, the distance Δd also shrinks, but their ratio Δd/Δt approaches a definite value — the instantaneous speed.

instantaneous speed = limit (Δd / Δt) as Δt → 0

On a position-time graph, the instantaneous speed at any moment equals the magnitude of the slope of the tangent line to the curve at that moment. A tangent line just touches the curve at one point without crossing it. Steep tangent = high speed; shallow tangent = low speed; horizontal tangent = zero speed, meaning the object is momentarily stopped.

7.4 Velocity as Speed with Direction

Velocity is the vector version of speed: it encodes both how fast an object is moving and in which direction. Two objects with the same speed but opposite directions have the same speed but different velocities.

In one dimension, direction is encoded by sign. Positive velocity means motion in the positive direction; negative velocity means motion in the negative direction. In two or three dimensions, velocity is a vector with components along each axis.

7.5 Average Velocity

Average velocity is displacement divided by elapsed time. Unlike average speed, which uses distance, average velocity uses displacement and therefore has direction.

v̅ = Δx / Δt = (x_f − x_i) / (t_f − t_i)

Average velocity can be zero even when the object has been moving: if the displacement is zero, meaning the object starts and finishes at the same place, the average velocity is zero. Average speed over the same trip would be positive.

7.6 Instantaneous Velocity

Instantaneous velocity is the velocity at one specific moment in time. It is the limit of average velocity as the time interval shrinks to zero. On a position-time graph, instantaneous velocity at any moment equals the slope of the tangent line at that point — including its sign.

v_inst = limit (Δx / Δt) as Δt → 0

The slope of the tangent to an x-t curve gives both the magnitude of instantaneous velocity, meaning the steepness, and its direction, meaning the sign of the slope. A positive slope means positive velocity; a negative slope means negative velocity; a zero slope means instantaneous velocity = 0, so the object is momentarily at rest even if it is about to start moving again.

7.7 Positive and Negative Velocity

The sign of velocity tells you the direction of motion relative to the chosen positive direction, not whether the object is speeding up or slowing down. A negative velocity simply means the object is moving in the negative direction.

An object with negative velocity is not necessarily decelerating. An object moving at −10 m/s and then at −15 m/s has increased its speed — it is moving faster in the negative direction. The velocity became “more negative,” which means the speed increased.

7.8 What Acceleration Means

Acceleration is any change in velocity. This is the single most important conceptual statement in this chapter. Students often think of acceleration as synonymous with “speeding up,” but this is incomplete and leads to systematic errors.

An object accelerates whenever its velocity changes — in magnitude, in direction, or in both. This means:

• Speeding up in the positive direction: positive acceleration.
• Slowing down in the positive direction: negative acceleration, also called deceleration.
• Speeding up in the negative direction: negative acceleration, because the object is already moving negatively and getting faster at it.
• Slowing down in the negative direction: positive acceleration, because the velocity is becoming less negative.
• Turning a corner at constant speed: acceleration, because the direction of velocity is changing.

7.9 Average Acceleration

Average acceleration is the change in velocity divided by the elapsed time. It is a vector quantity: it has both magnitude and direction.

a̅ = Δv / Δt = (v_f − v_i) / (t_f − t_i)

7.10 Instantaneous Acceleration

Instantaneous acceleration is the acceleration at one specific moment in time: the limit of average acceleration as the time interval approaches zero. On a velocity-time graph, instantaneous acceleration at any moment equals the slope of the tangent to the v-t curve at that point.

a_inst = limit (Δv / Δt) as Δt → 0

For constant acceleration, the most common situation in this book, average and instantaneous acceleration are identical — the acceleration does not change over time.

7.11 Positive vs. Negative Acceleration

The sign of acceleration tells you the direction of the acceleration vector, which determines how the velocity is changing. To figure out whether an object is speeding up or slowing down, compare the signs of velocity and acceleration.

7.12 Constant vs. Changing Acceleration

Constant acceleration means the acceleration has the same magnitude and direction throughout the motion. The velocity changes by the same amount in each equal time interval. This is the most important and most common case in introductory physics, and it is the only case for which the kinematic equations of Chapter 8 directly apply.

Changing acceleration means the acceleration itself varies over time. This occurs when the net force on the object changes — for example, in a car where the engine output varies, or in fluid drag where the force depends on speed. Treating changing acceleration with constant-acceleration equations gives wrong answers.

7.13 Velocity-Time Graphs

A velocity-time (v-t) graph plots velocity on the vertical axis and time on the horizontal axis. It is the most powerful tool for analyzing one-dimensional motion because both velocity, read directly from the graph, and acceleration and displacement, read from slope and area, are encoded in it.

Reading a v-t Graph

• The value of v at any time t is read directly from the graph.
• The slope of the v-t graph equals acceleration: steep positive slope = large positive acceleration; negative slope = negative acceleration; flat slope = constant velocity.
• The area under the v-t graph between the curve and the time axis equals displacement over that time interval.
• When the v-t curve crosses v = 0, the object has momentarily stopped and is about to reverse direction.
• A v-t graph above v = 0 means positive velocity; below v = 0 means negative velocity.

slope of v-t graph = Δv / Δt = acceleration (m/s²)

area under v-t graph = displacement (m)

7.14 Acceleration-Time Graphs

An acceleration-time (a-t) graph plots acceleration on the vertical axis and time on the horizontal axis. For constant acceleration — the most common case in this book — the a-t graph is simply a horizontal line.

• A horizontal line above a = 0: constant positive acceleration.
• A horizontal line below a = 0: constant negative acceleration.
• A line at a = 0: zero acceleration; constant velocity.
• The area under an a-t graph equals the change in velocity over that time interval: Δv = aΔt.

area under a-t graph = Δv = change in velocity (m/s)

The a-t, v-t, and x-t graphs form a hierarchy: the slope of the x-t graph gives v; the slope of the v-t graph gives a. Reading down this hierarchy involves taking slopes; reading upward involves computing areas. Keeping this three-graph structure in mind will support your understanding through Chapter 9.

7.15 Common Misconceptions About Acceleration

Chapter Summary

Speed and velocity both describe how rapidly position changes, but velocity also encodes direction. Acceleration is any change in velocity — in magnitude, direction, or both.

• Average speed = total distance / total time (scalar, always ≥ 0).
• Average velocity = displacement / elapsed time (vector, can be negative or zero).
• Instantaneous speed/velocity = the limit of average speed/velocity as Δt → 0; equals the slope of the tangent to the x-t curve.
• Average acceleration = Δv / Δt (vector); SI unit m/s².
• If v and a have the same sign, the object speeds up. If they have opposite signs, the object slows down.
• Constant acceleration: v-t graph is a straight line; a-t graph is a horizontal line.
• v-t graph: slope = acceleration; area under curve = displacement.
• a-t graph: horizontal line for constant acceleration; area = change in velocity.
• Acceleration ≠ speeding up. Zero velocity ≠ zero acceleration. Negative acceleration ≠ slowing down; always compare signs of v and a.

Key Terms

speed

The scalar magnitude of velocity; total distance divided by total time; always ≥ 0.

average speed

Total distance traveled divided by total elapsed time; a scalar quantity.

instantaneous speed

The speed at one specific instant; the magnitude of the instantaneous velocity; equals the magnitude of the slope of the tangent to the x-t graph.

velocity

A vector quantity describing the rate of change of position; displacement divided by time; includes both magnitude and direction.

average velocity

Displacement divided by elapsed time: v̅ = Δx / Δt; a vector quantity.

instantaneous velocity

Velocity at a specific instant; the limit of Δx/Δt as Δt → 0; equals the slope of the tangent to the x-t graph, including sign.

acceleration

The rate of change of velocity: a = Δv / Δt; a vector quantity; any change in velocity, magnitude, or direction.

average acceleration

Change in velocity divided by elapsed time: a̅ = Δv / Δt; SI unit m/s².

instantaneous acceleration

Acceleration at a specific instant; equals the slope of the tangent to the v-t graph.

constant acceleration

Acceleration that does not change over time; produces a straight line on a v-t graph and a horizontal line on an a-t graph.

deceleration

Informal term for acceleration directed opposite to the velocity; not a separate physical quantity from acceleration.

free fall

Motion under gravity alone, with no air resistance; produces constant downward acceleration of g = 9.81 m/s².

velocity-time graph

A graph of velocity vs. time; slope = acceleration; area under curve = displacement.

acceleration-time graph

A graph of acceleration vs. time; horizontal line for constant acceleration; area = change in velocity.

Concept Review Questions

Answer in complete sentences. No calculation required.

1. What is the difference between average speed and average velocity? Give a situation where they are equal and one where they are different.
2. Can an object have zero average velocity but non-zero average speed? Explain with an example.
3. A car travels north at 40 km/h and another travels south at 40 km/h. Do they have the same speed? The same velocity? Explain.
4. What does it mean for an object to accelerate? Give three physically different examples of acceleration.
5. A ball is thrown straight up. At the peak of its trajectory, what is its velocity? What is its acceleration? Are either of these zero?
6. An object moves with velocity −12 m/s and acceleration −3 m/s². Is it speeding up or slowing down? Explain.
7. What does the slope of a v-t graph represent? What does the area under a v-t graph represent?
8. Why does a car rounding a curve at constant speed have acceleration? What direction does the acceleration point?
9. Explain the difference between instantaneous velocity and average velocity. Under what conditions are they equal?
10. A v-t graph shows a horizontal line at v = −6 m/s for 5 seconds. What is the acceleration? What is the displacement over those 5 seconds?

Speed and Velocity Problems

Show all work. Report speed as positive; report velocity with sign.

11. A sprinter runs 100 m in 9.85 s. What is the average speed in m/s? In km/h?
12. A car travels from city A to city B, 240 km apart, in 2.5 h. Then it returns in 3.0 h. Find: (a) average speed for each leg, (b) average velocity for the whole trip, and (c) average speed for the whole trip.
13. An object starts at x = +8.0 m and moves to x = −6.0 m in 7.0 s. Find: (a) displacement, (b) average velocity, and (c) distance traveled, assuming no reversal.
14. A swimmer swims 50.0 m at 1.20 m/s, turns, and swims 50.0 m back at 1.50 m/s. Find: (a) time for each leg, (b) total distance, (c) total displacement, (d) average speed for the round trip, and (e) average velocity for the round trip.
15. An x-t graph has a straight line from (0 s, −3.0 m) to (6.0 s, +9.0 m). Find the average velocity over this interval.
16. At t = 2.0 s, a car is at x = +10.0 m. At t = 5.0 s, it is at x = +25.0 m. Find the average velocity.

Acceleration Problems

17. A car accelerates from rest to 28.0 m/s in 8.0 s. Find the average acceleration.
18. A bus brakes from 72.0 km/h to 0 in 6.0 s. Find the average acceleration in m/s².
19. A ball thrown upward has v_i = +15.0 m/s and reaches v_f = 0 m/s at the peak after 1.53 s. Find: (a) average acceleration, (b) its direction, and (c) whether this is consistent with free fall, g = 9.81 m/s².
20. An object has v_i = −10.0 m/s and v_f = −25.0 m/s after 5.0 s. Find: (a) acceleration, and (b) whether the object is speeding up or slowing down.
21. A train’s velocity changes from +20.0 m/s to −5.0 m/s in 10.0 s. Find the average acceleration.
22. An object has a = +3.0 m/s². Starting from v_i = +2.0 m/s, what is the velocity after 4.0 s?

Motion Graph Interpretation Exercises

Use the following v-t graph description: the graph has three segments. Segment A: v increases linearly from 0 to +12 m/s over 0–4 s. Segment B: v stays constant at +12 m/s from 4–7 s. Segment C: v decreases linearly from +12 m/s to 0 over 7–10 s.

23. Find the acceleration during each segment: A, B, and C.
24. Find the displacement during each segment.
25. Find the total displacement from t = 0 to t = 10 s.
26. Sketch the corresponding a-t graph for 0–10 s.
27. Sketch the corresponding x-t graph for 0–10 s, assuming x = 0 at t = 0. What shape is each segment?
28. A v-t graph shows a straight line from (0 s, +8 m/s) to (5 s, −2 m/s). Find: (a) acceleration, (b) the time at which the object momentarily stops, and (c) displacement from t = 0 to t = 5 s.

Mixed Kinematics Practice

29. A car is at rest. It accelerates at +4.0 m/s² for 5.0 s. Find: (a) final velocity, (b) the area under the v-t graph for this interval, and (c) what that area represents.
30. A ball rolls with v_i = +6.0 m/s and a = −2.0 m/s² due to friction. Find: (a) when it stops, (b) its velocity at t = 1.0 s, t = 2.0 s, and t = 4.0 s, and (c) whether it is still decelerating at t = 4.0 s.
31. Two cars start from the same point at the same time. Car A has v = +15 m/s constant. Car B starts at rest and has a = +3.0 m/s². Find: (a) after how many seconds Car B reaches the same speed as Car A, (b) rough v-t graphs for both cars on the same axes, and (c) which car has traveled farther at t = 5.0 s.
32. Sketch an x-t, v-t, and a-t graph for: “A ball is dropped from rest and falls for 3 seconds under constant gravitational acceleration, with downward defined as negative.”

Chapter 7 Checkpoint