Describe straight-line motion with position, displacement, velocity and acceleration as one-dimensional vectors measured from a frame of reference

Learning goal: Describe the motion of an object from a stated frame of reference using distance, position, displacement, speed, velocity, acceleration and time; tell vector quantities from scalar quantities; calculate displacement, average velocity and average acceleration in one dimension with correct signs and units; and decide from measured data whether an object moves with constant or changing velocity.

Motion depends on who is watching. A passenger reading a book on a train is at rest relative to the seat and moving at 30 m/s relative to a farmer beside the track. Neither description is wrong. Motion is always measured from a frame of reference: a chosen point taken as the origin, a chosen positive direction, and a clock. There is no special motionless frame from which every motion can be judged; the farmer is moving relative to the train just as surely as the train is moving relative to the farmer. In this course the relative nature of motion is treated conceptually: you must always state which frame you are using, and you should expect a description made from a different frame to give different positions and velocities for the same object. Frames that are themselves accelerating are left for a later physics course.

Scalars and vectors. A scalar is a quantity with size only: 70 m of distance, 1.4 m/s of speed, 50 s of time, 2.0 kg of mass. A vector has size and direction: a position of +2.0 m, a displacement of −5.0 m, a velocity of −2.0 m/s, an acceleration of +2.5 m/s². In one dimension, direction is carried by a sign. You choose which direction is positive (east, up, or to the right are common choices), and then everything pointing the other way is negative. A sign is not a judgment about "good" or "bad" motion; it is a direction.

Position and displacement. Position, x, is where an object is relative to the origin. Displacement is the change in position: Δx = x_f − x_i, final position minus initial position. Displacement is a vector, so it can be negative. Distance is the total length of the path traveled and is a scalar that is never negative. The two are often different. Walk 40 m east and then 30 m west: the distance traveled is 70 m, but the displacement is +10 m (taking east as positive), because you ended only 10 m east of where you started. Walk around a full running track and your distance is 400 m while your displacement is zero. Displacement asks only "where did you end up compared with where you began?"

Speed and velocity. Velocity is the rate at which position changes and is a vector. Average velocity is displacement divided by elapsed time: v_avg = (x_f − x_i)/(t_f − t_i). Speed is the rate at which distance is covered and is never negative (zero for an object at rest); average speed is distance divided by time. For the 40 m east, 30 m west walk taking 50 s, average speed is 70 m ÷ 50 s = 1.4 m/s, while average velocity is +10 m ÷ 50 s = +0.2 m/s. The speedometer of a car reads instantaneous speed, the speed at one clock reading. Instantaneous velocity adds the direction. When an object speeds up, slows down or turns around, its velocity changes from instant to instant, and the average over an interval need not equal the value at any particular moment inside it.

Constant velocity. An object moving with constant velocity has the same displacement in every equal time interval. Suppose a cart is at 0, 0.4, 0.8 and 1.2 m at 0, 1, 2 and 3 s. Each second the displacement is +0.4 m, so the average velocity in every interval is +0.4 m/s and the data are consistent with a constant velocity of +0.4 m/s (readings taken once a second cannot show what happened between them). Now suppose a second cart is at 0, 0.2, 0.8 and 1.8 m at the same times. The displacements are 0.2, 0.6 and 1.0 m in successive seconds: they grow, so the velocity is changing. This test, comparing displacement across equal intervals, is how you decide whether data are consistent with constant velocity: unequal displacements rule it out, while equal ones support it without proving what happened between readings.

Acceleration. Acceleration is the rate at which velocity changes and is also a vector: a_avg = (v_f − v_i)/(t_f − t_i), measured in meters per second per second, written m/s². A car going from 5 m/s to 20 m/s in 6 s has an average acceleration of 15 m/s ÷ 6 s = 2.5 m/s²: if the acceleration is constant, its velocity grows by 2.5 m/s every second (otherwise 2.5 m/s per second is the average rate). An object that is not accelerating is either standing still or moving with constant velocity, meaning constant speed and constant direction. Constant acceleration means the velocity changes by the same amount in every equal time interval.

A worked model: the sign of acceleration. A cart moving in the negative direction at −2 m/s speeds up to −8 m/s over 3 s. Its acceleration is (−8 − (−2)) m/s ÷ 3 s = −6 m/s ÷ 3 s = −2 m/s². The acceleration is negative, yet the cart is going faster. This is the misconception to root out: negative acceleration does not mean slowing down. It means the velocity is changing toward the negative direction. Whether the speed rises or falls depends on whether the acceleration and the velocity point the same way. "Deceleration" is an ambiguous word; use it, if at all, only when an object is actually slowing, and never treat it as a synonym for negative acceleration. Compare a cyclist moving in the positive direction at 12 m/s who brakes to rest in 4 s: a = (0 − 12) m/s ÷ 4 s = −3 m/s², negative and slowing, because here the acceleration opposes the velocity.

Motion diagrams. A motion diagram is a row of dots showing an object's position at equal time intervals, with an arrow for velocity at each dot. Equally spaced dots mean constant velocity; dots that spread out mean the object is speeding up; dots that bunch up mean it is slowing. In this course motion diagrams show position and velocity; drawing acceleration on them is reserved for a later physics course.

Investigation: a speed detection method. Materials: a tape measure, chalk or tape for marks, two stopwatches, a battery-powered toy car (or a walking classmate), a straight hallway. Steps: mark an origin and a positive direction; place marks at 0, 5, 10, 15 and 20 m; start the car from behind the origin so it is already moving when it passes 0; two timers record the clock reading as the car passes each mark; repeat three runs. Record a table of position against time, and compute the displacement and average velocity for each 5 m interval. If the four interval velocities agree within the scatter of your timing, the car moves with constant velocity; if they steadily change, its velocity is changing. A result that supports constant velocity: interval velocities such as 1.6, 1.6, 1.5 and 1.6 m/s. What this method cannot give you is instantaneous velocity at a single mark, only an average over each interval, and human reaction time of a few tenths of a second limits its precision; a motion sensor or frame-by-frame video reduces that limit.

Worked example 1 — displacement and average velocity with signs. A remote-controlled car starts at position x_i = +2.0 m on a line where right is positive and finishes at x_f = −3.0 m after 2.5 s. Displacement: Δx = x_f − x_i = −3.0 m − (+2.0 m) = −5.0 m. The negative sign says the net change in position was 5.0 m to the left. Average velocity: v_avg = Δx ÷ Δt = −5.0 m ÷ 2.5 s = −2.0 m/s, so on average the car moved 2.0 m to the left each second. If the car had first gone 1.0 m to the right before turning back, its distance would be 1.0 m + 6.0 m = 7.0 m and its average speed 7.0 m ÷ 2.5 s = 2.8 m/s, larger than the size of the average velocity because the path doubled back.

Worked example 2 — is the velocity constant? A ball rolling on a floor passes marks at these positions and clock readings: 0 m at 0 s, 1.5 m at 1 s, 3.0 m at 2 s, 4.5 m at 3 s. Displacements in successive one-second intervals: 1.5, 1.5, 1.5 m, all equal, so the data are consistent with a constant velocity of +1.5 m/s and zero acceleration (the marks cannot show what happened between readings). A second ball reads 0, 1.0, 3.0, 6.0 m at the same times. Its interval displacements are 1.0, 2.0 and 3.0 m, increasing by 1.0 m each second, so its velocity is changing: 1.0 m/s, then 2.0 m/s, then 3.0 m/s on average across each interval. Because the change in interval-average velocity is the same each second, the data are consistent with a constant acceleration of about 1.0 m/s².

Worked example 3 — acceleration from a change in speed. A train moving in the positive direction increases its velocity from 5.0 m/s to 20.0 m/s in 6.0 s. Average acceleration: a = (20.0 − 5.0) m/s ÷ 6.0 s = 15.0 m/s ÷ 6.0 s = 2.5 m/s². Units check: (m/s) ÷ s = m/s². Now the train brakes from 20.0 m/s to 8.0 m/s in 4.0 s: a = (8.0 − 20.0) m/s ÷ 4.0 s = −3.0 m/s². The acceleration is negative because velocity is changing toward the negative direction, and since the train's velocity is positive, it is slowing. Finally, a cart moving at −2.0 m/s reaches −8.0 m/s in 3.0 s: a = (−8.0 − (−2.0)) ÷ 3.0 = −2.0 m/s², negative, and this cart is speeding up. Same sign, opposite outcome, because what matters is whether acceleration and velocity share a direction.

Worked example 4 — frame of reference. A person walks toward the front of a train at 1.5 m/s relative to the train while the train moves at 30 m/s relative to the ground, both in the positive direction. Relative to the train the walker moves slowly forward, at 1.5 m/s. Relative to the ground she moves forward too, and a little faster than the train itself, because she is carried along by the train and is also walking the same way. Relative to the walker, the train's seats drift slowly backward. Each description is correct in its own frame, and none is "the real" velocity. In this course relative motion is reasoned about in words like these: which way, and faster or slower than what. Calculating velocities between frames is left for Physics.

Representation task: a motion record with signs. Choose a positive direction and an origin. Write a short story of a bicycle ride of at least four segments (for example: ride 60 m east in 20 s, stop for 10 s, ride 90 m west in 30 s, ride 30 m east in 15 s). For each segment record the initial position, final position, displacement, elapsed time and average velocity in a table with units. Draw a motion diagram for the whole trip with dots at equal time intervals. Then compute the trip's total distance, total displacement, average speed and average velocity, and write two sentences explaining why the last two are different. Finally list every quantity in your table and label each as scalar or vector.

Independent performance: design and run a speed measurement. Plan, carry out and report a method for measuring the speed of a toy car, a rolling ball or a walking person along a straight path of at least 10 m, using marked positions and timers or a video with a known frame rate. Success criteria: the frame of reference (origin and positive direction) is stated before any measurement; the data table has at least five position-time pairs from at least two runs; displacement and average velocity are calculated for every interval with units and correct signs; the report states, with evidence from the interval velocities, whether the object moved with constant or changing velocity; the report explains what the method measures (average velocity over an interval) and what it cannot measure (instantaneous velocity at a point); and one source of measurement error is named along with how a motion sensor or video would reduce it.