Unit 2 is about describing straight-line motion precisely, using position, displacement, velocity, and the graphs that connect them. By the end, you should be able to calculate distance, displacement, speed, and velocity, and read slope and area on position-time and velocity-time graphs.
Read the overview first, then work through each topic note and answer its checkpoint before you look at the answer.
This unit builds the vocabulary and graph skills that the rest of mechanics depends on. Everything here stays on a single line, so direction comes down to a plus or minus sign. Later units on forces, energy, and momentum assume you can read these quantities and graphs quickly.
Honors Physics Unit 2 covers motion in one dimension. You start with reference frames and relative motion, then separate distance (total path length, a scalar) from displacement (change in position, a vector). Next you compare speed and velocity, including averages over a time interval and instantaneous values at a single moment. The unit finishes with motion graphs. On a position-time graph, the slope is velocity. On a velocity-time graph, the slope is acceleration and the signed area between the line and the time axis is displacement.
A passenger on a train moving at 20 m/s is at rest relative to the train but moving at 20 m/s relative to the platform. In one dimension, subtract velocities measured in the same frame to find one object's velocity relative to the other. Always state your origin and positive direction before you calculate anything.
Distance and speed are scalars, so they are never negative and ignore direction. Displacement and velocity are vectors. The sign of displacement gives the net direction, while the sign of velocity gives the direction of motion. Distance equals the size of displacement, and average speed equals the size of average velocity, when the object does not reverse direction.
The slope of a position-time graph gives velocity, and the slope of a velocity-time graph gives acceleration. The signed area between a velocity-time graph and the time axis gives displacement. Once you know which feature means what, you can switch between the two graph types.
Each quantity in this unit describes how the one before it changes over time. That chain explains why slopes move you from position to velocity to acceleration, and why area moves you back from velocity to displacement. Most of the unit's skills come from applying this one relationship.
Every description of motion depends on a reference frame, which means an origin, a positive direction, and an observer.
Speed tells you how fast something moves.
A position-time graph puts position on the vertical axis and time on the horizontal axis.
A velocity-time graph puts velocity on the vertical axis.
Every description of motion depends on a reference frame, which means an origin, a positive direction, and an observer. If you change the observer, the motion changes too. A passenger sitting on a train moving at 20 m/s is at rest relative to the train but moving at 20 m/s relative to the platform. In one dimension, relative velocities add with signs. If the passenger walks toward the front at 1.5 m/s, someone on the platform sees m/s. If the passenger walks toward the back, the platform observer sees m/s.
Once you have a frame, you can give an object's position and describe how it moved. Distance is the total path length. It is a scalar and is never negative. Displacement is the change in position, . It is a vector, and its sign gives the direction.
Worked example. A hiker walks 2 km north, then 1.5 km south, then 3 km north. Take north as positive. For distance, add the size of every leg: km. For displacement, add the signed legs: km, so the hiker ends up 3.5 km north.
Distance is always greater than or equal to the size of the displacement. The two are equal only when the object never reverses. A runner who travels 200 m along a straight path and returns to the start has covered 400 m of distance but has zero displacement.
Check yourself: A car drives 5 km east, then 8 km west. What are its distance and displacement? (Answer: The distance is 13 km. The displacement is 3 km west, or -3 km if east is positive.)
Speed tells you how fast something moves. Velocity tells you how fast and in which direction. Their averages use different quantities on top of the fraction:
and
Worked example. You jog 600 m east in 200 s, then walk 200 m back west in another 200 s. The total distance is 800 m and the total time is 400 s, so your average speed is m/s. Your displacement is m, so your average velocity is m/s, or 1.0 m/s east. The two answers differ because you turned around.
Average values describe a whole time interval. Instantaneous values describe a single moment. A car's speedometer shows instantaneous speed, but it can't show direction, so it doesn't show velocity. Instantaneous speed is the size of instantaneous velocity, so a velocity of m/s means a speed of 4 m/s.
Velocity changes whenever the speed changes, the direction changes, or both. A ball tossed straight up has zero velocity for an instant at the top, but it doesn't stay there. It starts falling right away because gravity keeps changing its velocity.
Check yourself: A swimmer swims one 50 m length of a pool and back in 80 s. What are the average speed and average velocity? (Answer: The average speed is 100 m / 80 s = 1.25 m/s. The average velocity is 0 m/s because the displacement is zero.)
| Feature | Speed | Velocity |
|---|---|---|
| Scalar or vector | Scalar | Vector |
| Average uses | Total distance | Displacement |
| Can it be negative? | No | Yes, the sign shows direction |
| Average for a round trip | Greater than zero | Zero |
| What a speedometer shows | Yes, instantaneous speed | No, it shows no direction |
A position-time graph puts position on the vertical axis and time on the horizontal axis. The slope of the graph is the velocity.
A straight line means constant velocity. An upward slope means the object moves in the positive direction, a downward slope means it moves in the negative direction, and a horizontal line means it is at rest. A steeper line means a faster speed. If the graph dips below the time axis, the object is on the negative side of the origin. It does not mean the object has a negative distance.
A curved line means the velocity is changing, so the object is accelerating. When the magnitude of the slope increases, the object speeds up. When the magnitude decreases, the object slows down.
To find average velocity over an interval, take the slope of the secant line between two points on the graph. If an object is at 2 m at s and at 10 m at s, then m/s.
To find instantaneous velocity on a curve, draw the tangent line at that moment and calculate its slope. Use two points on the tangent line rather than two arbitrary points on the curve, and pick them far apart so your reading is more accurate. Where the tangent is horizontal, the velocity is zero, but the object reverses direction only if its velocity changes sign.
Check yourself: A position-time graph is a horizontal line at x = 6 m from t = 0 to t = 4 s. What is the object's velocity? (Answer: It is zero. The object sits at rest 6 m from the origin the whole time.)
A velocity-time graph puts velocity on the vertical axis. You can read three things from it. The value at any moment is the instantaneous velocity. The slope gives acceleration. Over an interval, average acceleration is ; at a moment, instantaneous acceleration is the tangent slope. The signed area between the line and the time axis is displacement.
Area above the axis counts as positive displacement, and area below the axis counts as negative displacement. Add the signed areas to get net displacement. Add the sizes of the areas, ignoring sign, to get total distance. Where the line crosses the time axis, the velocity changes sign, which means the object turned around.
The object speeds up when its velocity and acceleration have the same sign. It slows down when they have opposite signs. On the graph, that means the object speeds up when the line moves away from the time axis and slows down when the line moves toward it.
Worked example. A car starts from rest and speeds up steadily to 10 m/s in 4 s, then cruises at 10 m/s for 3 s. During the first 4 s, the slope is m/s². The area under that part is a triangle, m. The cruising part is a rectangle, m. The total displacement is m, so the average velocity over 7 s is m/s.
Check yourself: A velocity-time graph shows a triangle of area 12 m above the axis followed by a triangle of area 4 m below it. What are the net displacement and the total distance? (Answer: The net displacement is +8 m and the total distance is 16 m.)
| Feature | Position-time graph | Velocity-time graph |
|---|---|---|
| Value you read off the graph | Position | Velocity |
| Slope means | Velocity | Acceleration |
| Area under the line means | Nothing useful in this unit | Displacement |
| Horizontal line means | At rest | Constant velocity |
| Line crosses the time axis | Object passes the origin | Object reverses direction |
| Term | Definition |
|---|---|
| Kinematics | The part of physics that describes motion using position, velocity, and acceleration without asking what forces cause it. |
| Reference frame | The observer's viewpoint and coordinate system, including an origin and a positive direction, used to measure motion. |
| Scalar | A quantity with size only, such as distance or speed. |
| Vector | A quantity with both size and direction, such as displacement, velocity, or acceleration. |
| Displacement | The change in an object's position, found as final position minus initial position. |
| Average velocity | Displacement divided by the time interval, equal to the slope of the secant line on a position-time graph. |
| Instantaneous velocity | The velocity at a single moment, equal to the slope of the tangent line on a position-time graph. |
| Acceleration | The rate at which velocity changes, equal to the slope of a velocity-time graph. |
| Relative velocity | The velocity of an object relative to an observer. In one dimension, subtract the observer's velocity from the object's velocity when both are measured in the same frame. |
| Area under the curve | On a velocity-time graph, the signed area between the line and the time axis, which equals displacement. |
The sign of acceleration only gives its direction compared with your chosen positive direction. An object with negative velocity and negative acceleration is speeding up in the negative direction. Compare the signs instead: the object speeds up when velocity and acceleration have the same sign and slows down when they have opposite signs.
A hump on a position-time graph does not mean the object went over a hill. It means the object moved in the positive direction, stopped for an instant, and came back. Before you interpret any graph, read the axis labels and remember that the horizontal axis is time, not horizontal location.
Average speed uses total distance, while average velocity uses displacement. These match only if the object never turns around. If you jog 600 m east and walk 200 m back west in 400 s, your average speed is 2.0 m/s, but your average velocity is only 1.0 m/s east.
At the top of a vertical toss, the ball's velocity is zero for an instant, but its acceleration is still 9.8 m/s² downward. That is why it starts falling right away. On a velocity-time graph, the line can cross zero while its slope stays the same.