Imagine a test track with three screens beside it. The first shows where the car is, the second its speedometer with a sign for direction, and the third a meter for its acceleration. Each screen draws a graph as the car runs. The three graphs look nothing alike, yet they describe one car. An engineer who sees only one screen can still tell you what the other two show.
Three views of one motion
Every motion along a line has three graphs: a position-time graph, a velocity-time graph and an acceleration-time graph. They are not independent. At each instant, the slope of the x-t graph must equal the height of the v-t graph. The slope of the v-t graph must equal the height of the a-t graph. So the three graphs must agree instant by instant, like three witnesses of one event.
Play the three chips one after another and read each column of graphs from the top.
- Constant velocity, \(10\,\text{m/s}\). The x-t graph is a straight line. The v-t graph is a flat line at \(10\,\text{m/s}\). The a-t graph lies on the time axis.
- Speeding up from rest at \(5\,\text{m/s}^2\). The x-t graph bends upward. The v-t graph is a rising straight line. The a-t graph is a flat line at \(+5\,\text{m/s}^2\).
- Slowing down from \(20\,\text{m/s}\) at \(5\,\text{m/s}^2\). The x-t graph bends downward and flattens. The v-t graph is a falling straight line that reaches zero at \(4\,\text{s}\). The a-t graph is a flat line at \(-5\,\text{m/s}^2\).
Notice what happens to the shapes as you go down. A parabola on the x-t graph becomes a straight line on the v-t graph. A straight line becomes a flat line on the a-t graph. Each slope lowers the power of \(t\) by one. For constant acceleration, \(x\) has \(t^2\), \(v\) has \(t\), and \(a\) has no \(t\) at all.
Read the three graphs of one instant like three questions. Where is the body? The x-t graph. How fast and which way? The v-t graph. Is that answer changing, and which way? The a-t graph. Each graph answers the question about the one above it.
Features that must line up
Some features of one graph always sit at the same instant as a feature of another. Learning these pairs is the fastest way to check a sketch.
- A peak or a valley of the x-t graph is where the v-t graph crosses the time axis. At that instant \(v = 0\) and the body turns back.
- At the steepest point of the x-t graph its bend changes, from cup to dome or back. There the v-t graph is highest or lowest, and the a-t graph changes sign.
- A flat piece of the x-t graph means \(v = 0\) on the v-t graph for that whole stretch, and \(a = 0\) too.
The x-t graph of a particle has its highest point at \(t = 3\,\text{s}\). What does the v-t graph show at \(t = 3\,\text{s}\)?
The v-t graph of a car is a straight line that starts at \(+15\,\text{m/s}\) and falls to zero at \(t = 5\,\text{s}\). Which describes its x-t and a-t graphs from \(0\) to \(5\,\text{s}\)?
Sketching the other two
The ladder of the last lesson now becomes a method. Given the v-t graph, take its slopes to draw the a-t graph and its areas to draw the x-t graph. If you are given the x-t graph, take slopes twice. If you are given the a-t graph, take areas twice, and ask for the initial velocity and position. Work one straight or curved piece at a time.
A sketch is not a plot. It needs the right shape in each piece and the right values at the joins, not exact points everywhere. For each piece, decide three things: the sign of the height, whether the graph rises or falls, and whether it is straight or bent. Then fix the numbers at the ends of the piece.
A scooter starts from rest at \(x = 0\). Its v-t graph rises in a straight line to \(6\,\text{m/s}\) at \(3\,\text{s}\) and stays there until \(6\,\text{s}\). It then falls in a straight line to zero at \(8\,\text{s}\). Sketch its a-t and x-t graphs.
Three pieces: \(0\) to \(3\,\text{s}\), \(3\) to \(6\,\text{s}\), \(6\) to \(8\,\text{s}\). Each piece of the v-t graph is a straight line, so each has one constant slope.
Slopes give the a-t graph: \(6/3 = +2\), then \(0\), then \(-6/2 = -3\,\text{m/s}^2\). Areas give the displacement in each piece: \(\tfrac12 \times 3 \times 6 = 9\,\text{m}\), \(3 \times 6 = 18\,\text{m}\), \(\tfrac12 \times 2 \times 6 = 6\,\text{m}\).
The a-t graph is three flat steps: \(+2\), \(0\), \(-3\,\text{m/s}^2\). The x-t graph passes through \(0\), \(9\), \(27\) and \(33\,\text{m}\) at \(0\), \(3\), \(6\) and \(8\,\text{s}\).
Draw the x-t graph piece by piece. From \(0\) to \(3\,\text{s}\) it bends upward from a flat start. From \(3\) to \(6\,\text{s}\) it is a straight line of slope \(6\,\text{m/s}\). From \(6\) to \(8\,\text{s}\) it bends downward and ends flat at \(33\,\text{m}\). The joins are smooth, because the velocity never jumps.
Four ways to say one motion
A motion can be told in words, drawn as a graph, written as an equation, or shown as a picture. Each form makes some facts obvious and hides others. The block below shows one braking car in all four forms. Move between the tabs and find the same facts in each: the starting speed, the stopping time and the stopping distance.
A car passes a school gate at \(12\,\text{m/s}\). The driver brakes at once, with a steady deceleration of \(2\,\text{m/s}^2\). The car covers \(11\,\text{m}\) in the first second, \(9\,\text{m}\) in the next, then \(7\), \(5\), \(3\) and \(1\,\text{m}\). It stops \(6\,\text{s}\) later, \(36\,\text{m}\) past the gate.
For the braking car, \(x = 12t - t^2\). Using the equation or the graph, find the position of the car at \(t = 3\,\text{s}\), in metres.
A body starts from rest. Its a-t graph is a flat line at \(+4\,\text{m/s}^2\) from \(0\) to \(2\,\text{s}\) and zero after that. Which is its v-t graph?
A body moves with constant negative acceleration and positive initial velocity. Which statements about its graphs are true?
In a cricket broadcast, cameras around the ground record the position of the ball many times a second. Software joins these positions into a position-time record for each direction. The slope of that record, taken over a few frames, gives the velocity, and the screen shows the speed of the delivery in km/h. The same numbers, used again, predict the path of the ball for a leg-before-wicket decision.
- Copying the shape of one graph onto the next.The next graph down shows the slope, not the shape. A rising straight x-t line becomes a flat v-t line.
- Drawing the top of an x-t dome at the same time as the top of the v-t graph.The top of the x-t graph is where \(v = 0\). The v-t graph is highest where the x-t graph is steepest.
- Climbing from an a-t graph without a starting velocity.Areas give only changes. Ask for \(u\) and \(x_0\) before you draw the upper graphs.