Riya walks from her home to the shop at the end of her lane, \(120\,\text{m}\) away, at a steady pace. She spends a minute choosing biscuits. Then she remembers that her tea is on the stove and hurries home, faster than she came. Her whole trip can be drawn as three straight lines.
From a story to a graph
A story of a motion comes in stages: "she walked", "she waited", "she hurried back". Each stage becomes one piece of the graph. For each piece, decide three things from the words. Is the body moving forward, backward, or not at all? Is its speed steady or changing? Where does the stage start and end? Then draw the pieces one after another, each starting where the last one ended.
Riya walks \(120\,\text{m}\) to the shop at \(1.2\,\text{m/s}\), stays for \(60\,\text{s}\), and walks home at \(2\,\text{m/s}\). Take her home as the origin and the way to the shop as positive. Sketch her x-t graph.
Three stages: walking out, waiting, walking back. Each has a steady velocity, so each is a straight piece of the x-t graph.
Walking out takes \(120/1.2 = 100\,\text{s}\). Waiting lasts \(60\,\text{s}\). Walking back takes \(120/2 = 60\,\text{s}\).
The graph goes from \((0, 0)\) to \((100\,\text{s}, 120\,\text{m})\), flat to \((160\,\text{s}, 120\,\text{m})\), and down to \((220\,\text{s}, 0)\).
The first piece has slope \(+1.2\,\text{m/s}\), the second slope zero, the third slope \(-2\,\text{m/s}\). The last piece is steeper than the first, because Riya hurries back.
A school bus leaves a stop, speeds up, runs steadily, slows down and stops at the next stop. Put the pieces of its v-t graph in the order they appear.
From a graph to a story
Reading a graph backward into words is the real test of understanding. First read the label of the vertical axis, because the same shape means different things on different graphs. Then take the pieces one at a time. For each piece, say where the body is, which way it moves, and whether it is speeding up, slowing down or steady. Finally, join the pieces into one story that a friend could act out.
The v-t graph of a cyclist stays flat at \(4\,\text{m/s}\) for \(10\,\text{s}\). It then falls in a straight line to \(-2\,\text{m/s}\) over the next \(6\,\text{s}\). Tell the story of the ride in two or three sentences.
The cyclist rides forward at a steady \(4\,\text{m/s}\) for \(10\,\text{s}\). Then the cyclist brakes steadily and stops for an instant \(4\,\text{s}\) later. The cycle then rolls backward, perhaps down a slope, reaching \(2\,\text{m/s}\) backward at the end. The falling line crosses the axis at the turning point, where the ride changes direction.
Trap one: speed-time is not velocity-time
A speedometer shows speed, not velocity. Speed is the magnitude of the velocity, \(|v|\), so a speed-time graph can never go below the time axis. Where a v-t graph dips below the axis, the speed-time graph is its mirror image, folded up above the axis. The two graphs agree while the body moves forward and differ as soon as it moves backward.
The fold changes what the graph can tell you. The area under the speed-time graph is the distance, \(20 + 20 = 40\,\text{m}\), never the displacement. Worse, the slope of the speed-time graph is not the acceleration once the body moves backward. After the top, the speed-time graph rises with slope \(+10\,\text{m/s}^2\), but the acceleration of the ball is still \(-10\,\text{m/s}^2\). The slope of a speed-time graph gives only the rate at which the speed changes.
A ball thrown straight up has a V-shaped speed-time graph. During the fall, the graph rises with slope \(10\,\text{m/s}^2\). Taking upward as positive, what is the acceleration of the ball during the fall?
Trap two: the graphs of a projectile
A ball thrown at an angle, a projectile, has a path, and it also has graphs. The path is \(y\) against \(x\), a picture of where the ball has been in space. The graphs put time on the horizontal axis. Look at the four panels of the figure and name each one before you press play.
Two of the panels are parabolas that look alike. The path has metres along the bottom; the y-t graph has seconds. They look alike only because the horizontal velocity is constant, so \(x = 12t\) grows in step with time. The x-t graph is a straight line, because nothing accelerates the ball sideways. The \(v_y\)-t graph is a straight line of slope \(-10\,\text{m/s}^2\), exactly the v-t graph of a ball thrown straight up.
The speed of the projectile is the trap inside the trap. The vertical velocity falls to zero at the top, but the horizontal velocity stays at \(12\,\text{m/s}\). So the speed at the top is \(12\,\text{m/s}\), the smallest speed of the flight, not zero. The speed-time graph of a projectile is a smooth sag from \(20\,\text{m/s}\) down to \(12\,\text{m/s}\) and back.
For the projectile in the figure, find the speed of the ball at the top of its path, in m/s.
For a projectile launched from the ground, with air resistance neglected, what is the shape of the graph of horizontal position \(x\) against time?
Trap three: the curve is not the path
The first lesson of this chapter warned that an x-t graph is not a path. The projectile shows why the warning matters. Its y-t graph looks exactly like a path, so the eye takes it for one. A ball thrown straight up has the same y-t dome and yet moves along a single vertical line. Before you read any curve, read the horizontal axis. If it says \(t\), the curve is a graph of time, not a picture of space.
Which statements are true?
In 1786 the Scottish engineer William Playfair published an atlas of England's trade. Instead of long tables of numbers, he drew lines with the years running along the bottom and the money running up the side. Readers could see at a glance when trade rose, fell or stood still. Playfair's line graphs spread from economics into every science. Every x-t, v-t and a-t graph in this chapter keeps his rule: time along the bottom.
- Reading the slope of a speed-time graph as the acceleration after a reversal.The speed-time graph folds the v-t graph. Use the v-t graph for the acceleration.
- Taking the y-t graph of a projectile for its path.Check the horizontal axis. The path has \(x\) along the bottom; the graph has \(t\).
- Saying a projectile stops at the top.Only \(v_y\) is zero at the top. The speed there is the horizontal velocity \(u_x\).