Use speed-time records from a cart and ramp as evidence of net force

Learning goal: Measure a cart's speed at several times from a fixed reference point, plot a speed-time graph, and use the shape of the graph as evidence for whether the net force on the cart was zero or nonzero and in which direction it acted, including the effect of friction on different surfaces.

Why a graph is evidence. A force diagram predicts what should happen; a record of motion shows what did happen. If forces can act to change the motion of objects, then a changing speed or direction is evidence that a nonzero net force was acting, and a steady speed in a straight line is evidence that the forces were balanced. This lesson connects the two: read the motion record, infer the net force, then check the inference against a force diagram of the same moment. It builds on the previous lesson, which taught how forces are added to give a net force. A reminder with hypothetical numbers: forces are added taking direction into account, and the net force is the sum of all of them. If a cart on a level carpet has gravity of 5 N down, the floor's support of 5 N up and friction of 1.5 N backward, the vertical forces cancel and the net force is 1.5 N backward; if someone also pushed it forward with 1.5 N, the net force would be zero. Here that kind of sum is checked against the record of motion.

Reference point and speed. The motion of an object is always measured with respect to a reference point. For a cart on a track, choose a mark on the track, such as the bottom of the ramp, as position zero, and measure positions from there. Speed is the distance moved divided by the time taken. Because a cart's speed can change, what you measure over any interval is an average speed for that interval: a cart that passes two marks 0.60 m apart in 0.5 s had an average speed of 1.2 m/s between them. Short intervals give a closer estimate of the speed at one moment. A second observer moving alongside the track would record different positions and speeds, and neither record is wrong; they use different reference points, so a report must state which one it used.

Reading a speed-time graph. Put time on the horizontal axis and speed on the vertical axis. A horizontal line means the speed is not changing; if the direction is also constant, the forces on the object are balanced and the net force is zero. For an object moving along a straight line, as the cart does here, a line that rises means the object is speeding up, so a nonzero net force acts in the direction of motion, and a line that falls means the object is slowing down, so a nonzero net force acts opposite to the motion. (On a curved path a rising line shows only that part of the net force points forward.) The steeper the line, the faster the speed is changing. A speed-time graph does not show where the cart is or which way it is going; a separate position-time record, or a written note of direction, is needed for that.

The model data. The graph shows invented teaching values for one cart released from rest at the top of a gentle 1.35 m ramp that meets a level floor at the 3-second mark. These are not measurements from any real apparatus; they are round numbers chosen to make the reasoning clear. On the ramp, the speeds are 0, 0.3, 0.6 and 0.9 m/s at 0, 1, 2 and 3 seconds: the speed rises by 0.3 m/s every second, so the net force on the cart points down the slope and is roughly steady. Gravity pulls the cart down the slope harder than friction resists it. At 3 seconds the cart rolls onto the floor. On carpet, the speeds at 4, 5 and 6 seconds are 0.6, 0.3 and 0 m/s: the speed falls by 0.3 m/s each second until the cart stops. On the level floor gravity and the floor's support cancel, and the only horizontal force is friction, acting opposite to the motion, so the net force points backward and the cart slows. A word about which friction: the cart's wheels roll rather than slide, so this is not kinetic friction at the wheel-floor contact; it is friction in the axles and bearings plus rolling resistance, the resistance from wheels pressing into a surface, which is large on carpet because the wheels sink into the pile. All of it acts opposite to the cart's motion, exactly as kinetic friction does on a sliding block, and the force diagrams in this lesson label it simply "friction". (A block sliding down the same ramp would feel kinetic friction, but on a ramp this gentle most blocks will not start sliding at all.) On a smooth floor the same cart's speeds are 0.8, 0.7 and 0.6 m/s at 4, 5 and 6 seconds: still falling, but by only 0.1 m/s each second, because friction on the smooth surface is smaller, so the backward net force is smaller and the change in motion is smaller. Air drag also acts on the cart, opposite to its motion through the air, but at these speeds it is tiny compared with friction and is left out of the model. If a floor had no friction and no drag at all, the line after 3 seconds would be horizontal at 0.9 m/s, and the cart would keep rolling with nothing pushing it.

What the model cannot show. The straight lines are an idealization; a real cart's speed changes a little unevenly because of wheel wobble, air drag and bumps at the ramp's foot. The graph does not show the sizes of the forces, only whether the net force was zero, forward or backward and whether it was larger or smaller. And the graph alone cannot tell you why the net force was smaller on the smooth floor; that inference comes from the force diagram, in which the friction arrow is shorter.

A worked interpretation. Between 3 and 6 seconds on carpet, the cart's speed fell from 0.9 m/s to 0 m/s, a change of 0.9 m/s in 3 seconds, or 0.3 m/s each second. On the smooth floor over the same 3 seconds, the speed fell from 0.9 m/s to 0.6 m/s, a change of 0.3 m/s, or 0.1 m/s each second. The carpet changed the cart's motion three times as fast. Both records slope downward, so both show a backward net force; the steeper carpet line shows a larger one. The force diagrams agree: on both surfaces the vertical forces cancel and the only horizontal force is friction pointing backward, drawn longer for the carpet. Nothing in the record requires a forward force to keep the cart moving; the cart moves because it already had speed, and the only question is how quickly friction removes it.

Investigation: cart, ramp and two floors. Materials: a low-friction cart or toy car, a board propped as a ramp no higher than 15 cm at its top, masking tape marks every 20 cm along the ramp and along the floor as far as the cart rolls (several meters on a smooth floor), a stopwatch or a phone recording video, a strip of carpet or a towel taped flat. Steps: release the cart from rest at the same mark every time, without pushing; record the time at which the cart's front passes each tape mark, either with the stopwatch or by counting video frames at a known frame rate (a hand-held stopwatch cannot separate marks only a few tenths of a second apart, so with a stopwatch time every second mark, or use video, which is the better method); repeat three times on the bare floor and three times with the carpet strip laid after the ramp. For each interval between two timed marks, compute the average speed as the distance between those marks (0.20 m if every mark was timed, 0.40 m if every second mark was) divided by the time taken, and plot speed against the time at the middle of each interval. What to record: release mark, floor type, the times at every mark for every trial, and any trial where the cart left the tape line. A result that supports the claim: rising speeds on the ramp, falling speeds on the floor, and a faster fall on the carpet than on the bare floor. Safety: keep the floor area clear, catch the cart at the end of the track, and keep the ramp low enough that the cart cannot leave it.

Careful measurement. Three trials matter because timing by hand is uncertain by a tenth of a second or more. Report all trials rather than only the one that looks best, and use the middle value or the mean. Change one thing at a time: the same cart, the same ramp height, the same release mark, and only the floor surface changed between the two conditions. Then any difference in how quickly the speed falls can fairly be attributed to friction.

Worked example 1 — average speed from marks. A cart passes a mark at 0.40 m from the reference point when the stopwatch reads 1.0 s and a mark at 0.70 m when it reads 1.5 s. Distance moved: 0.70 minus 0.40 equals 0.30 m. Time taken: 1.5 minus 1.0 equals 0.5 s. Average speed over the interval: 0.30 m divided by 0.5 s equals 0.6 m/s. This is the average for the half second, not the speed at any single instant; a shorter interval would narrow that down.

Worked example 2 — reading net force from the graph. From 0 to 3 s the model line rises from 0 to 0.9 m/s: the cart speeds up, so the net force is nonzero and points along the motion, down the ramp. From 3 to 6 s on carpet the line falls from 0.9 to 0 m/s: the net force is nonzero and points against the motion. If a line were flat between 3 and 6 s, the speed would be unchanged and the horizontal forces balanced. The graph identifies the direction of the net force and compares its size between surfaces; it does not give the force in newtons.

Worked example 3 — two reference points. A student walks toward the front of a train at 1 m/s relative to the train while the train moves at 20 m/s relative to the ground. Relative to a passenger seated in the train, the student moves at 1 m/s. Relative to a person standing beside the track, the student moves at 21 m/s, because the walking and the train's motion are in the same direction and add. Both statements describe the same motion from different reference points; the seated passenger has not made an error by ignoring the train's speed, and neither has the observer on the ground.

Worked example 4 — deciding what changed. Two trials use the same cart and ramp, but in the second trial the cart stops sooner on the floor. A student claims the cart "had less energy" in trial two. Check the record first: if the ramp speeds in both trials match, the cart reached the floor with the same motion, and the difference lies on the floor, most likely a different surface, a bump or debris that increased friction. If the ramp speeds differ, look for a difference in release: a push, a different mark or a changed ramp height. The record narrows the explanation before any claim is made.

Representation task: plot and annotate. Using the model values from the lesson (ramp: 0, 0.3, 0.6, 0.9 m/s at 0 to 3 s; carpet: 0.6, 0.3, 0 m/s at 4 to 6 s; smooth floor: 0.8, 0.7, 0.6 m/s at 4 to 6 s), draw the speed-time graph on graph paper with labeled axes. Beside each segment write the direction of the net force (down the slope, backward, or zero) and draw a small force diagram of the cart at 4 s on carpet and at 4 s on the smooth floor, with the friction arrow drawn longer where friction is larger. On the ramp the support pushes at right angles to the sloping surface rather than straight up, so a ramp diagram is not a simple up-down and left-right one; for 1 s, write the direction of the net force in words instead of drawing it. Explain in two sentences what the graph shows that a single stopwatch reading could not.

Independent performance: the cart-and-ramp investigation. Carry out the investigation from the lesson, or analyze a teacher-provided timing record, and write a report. Success criteria: the reference point and direction are stated; a table lists every trial's times at every mark; average speeds are computed correctly for at least four intervals per trial with units; a speed-time graph shows both floor conditions; the conclusion states where the net force was zero, forward or backward, citing the slope of the graph as evidence; the report explains why the carpet run slowed faster using the words friction and net force, and says why the friction on a rolling cart is not kinetic friction at the wheels; and it names one limitation of the measurement, such as reaction time, and one way to reduce it.