A friend shows you a graph drawn from the GPS of a car. Time runs along the bottom and the position of the car along the side. The curve rises, flattens at the top and comes down again, like a small hill. "The car drove over a hill," your friend says. The road was perfectly flat and perfectly straight. So what did the car really do?
One curve, the whole journey
A body moves along a straight line. At every instant it has one position \(x\). Plot time \(t\) along the horizontal axis and position \(x\) along the vertical axis. Each instant gives one point, and the points together make a curve. This curve is the position-time graph, or the x-t graph. It stores the whole journey in one picture: where the body was at every instant.
The first reading is the height. The height of the curve above the time axis at some instant is the position at that instant. A point below the time axis means a negative position, on the negative side of the origin. The second reading is the slope. In the last chapter the slope of the tangent to the x-t graph proved to be the instantaneous velocity, \(v = dx/dt\). So the steepness of the curve tells how fast the body moves, and the direction of the tilt tells which way.
Watch the constant-velocity motion first. The car covers \(10\,\text{m}\) in every second, and its marks on the road are equally spaced. The graph is a straight line, and the slope reading stays at \(10\,\text{m/s}\). A straight x-t graph always means a constant velocity, because a straight line has the same slope everywhere.
On a position-time graph, the height of the curve at any instant is the position at that instant. The slope of the tangent at that instant is the velocity. A straight piece means constant velocity, a flat piece means rest, and a falling piece means motion in the negative direction.
Reading a journey piece by piece
The graph below belongs to an auto-rickshaw on a straight road. It is made of four straight pieces. Each straight piece has one slope, so each piece has one constant velocity. Read the pieces in order.
- From O to A, the position rises from \(0\) to \(100\,\text{m}\) in \(10\,\text{s}\). The slope is \(100/10 = +10\,\text{m/s}\). The auto moves forward.
- From A to B, the position stays at \(100\,\text{m}\) for \(10\,\text{s}\). The slope is zero. The auto is at rest, perhaps at a signal.
- From B to C, the position falls from \(100\,\text{m}\) to \(40\,\text{m}\) in \(10\,\text{s}\). The slope is \(-60/10 = -6\,\text{m/s}\). The auto moves backward, towards the origin.
- From C to D, the position rises from \(40\,\text{m}\) to \(60\,\text{m}\) in \(10\,\text{s}\). The slope is \(+2\,\text{m/s}\). The auto moves forward again, slowly.
A steep piece means a fast motion and a gentle piece means a slow one. The sign of the slope gives the direction. The height, on the other hand, says nothing about the direction of motion. At C the auto is on the positive side of the origin, yet it has just been moving in the negative direction.
In the auto-rickshaw graph, what is the velocity of the auto between B and C?
Use the auto-rickshaw graph. Find the average velocity of the auto over the whole trip, from O at \(0\) to D at \(40\,\text{s}\), in m/s.
The bend of the curve
Now switch the figure to the speeding-up chip and press play. The marks on the road spread apart. The graph is no longer straight: it starts flat and becomes steeper and steeper. The slope reading climbs from \(0\) to \(20\,\text{m/s}\). A slope that grows means a velocity that grows, which means a positive acceleration. The graph bends upward, like the inside of a cup.
Switch to the slowing-down chip. Now the graph starts steep and flattens out. The slope reading falls from \(20\,\text{m/s}\) to \(0\). A slope that falls means a velocity that falls, which means a negative acceleration. The graph bends downward, like an upturned cup or a dome.
Mathematicians call the cup shape concave up and the dome shape concave down. The acceleration is the rate of change of the slope, \(a = d^2x/dt^2\). So the bend of the x-t graph shows the sign of the acceleration at a glance, before any calculation.
The bend gives the sign of \(a\), not whether the body speeds up. Look at the top of a dome-shaped graph. Just after the top the slope is negative and growing more negative. The body moves backward and speeds up, yet the graph is still concave down. Speeding up needs \(v\) and \(a\) of the same sign, as the last chapter showed.
The x-t graph of a scooter bends upward like a cup throughout the motion. What can you say for certain?
The curve is not the road
Return to the graph shaped like a hill. Before you watch the car, decide what its path looks like.
A car moves on a straight, flat road. Its x-t graph rises, becomes flat at the top and falls again. What did the car do?
The car moves only along the road, a straight line. Its x-t graph is curved because the position changes with time in a curved way. The rise of the hill is the car moving forward and slowing down. The top is the turning point, the instant when the velocity is zero, at \(t = 2\,\text{s}\) and \(x = 12\,\text{m}\). The fall is the car returning and speeding up. The whole hill is concave down, because the acceleration is \(-6\,\text{m/s}^2\) throughout.
A position-time graph shows how one coordinate changes with time. Its shape is not the shape of the path. A body that moves along a straight line can have a curved x-t graph, and the direction of time is always to the right.
The x-t graph of a car on a straight road is \(x = 12t - 3t^2\) metres, from \(t = 0\) to \(t = 4\,\text{s}\). Which statements are true?
A student says that a curved x-t graph means the body moved along a curve. Explain in two or three sentences why this is wrong.
The vertical axis of an x-t graph is the position along one straight line, and the horizontal axis is time. The graph moves to the right only because time passes, not because the body moves sideways. A curve in the graph means that the position changes at a changing rate, so the velocity is changing. The body can still be on a straight line.
Indian Railways section controllers plan traffic on a time-distance chart. Time runs across the sheet and the stations of the section run down the side. Every train is a line on the chart. A steep line is a fast train, and a flat piece is a train waiting at a station. Two lines that cross are two trains passing each other. The controller reads speeds and waits from slopes, exactly as you read the auto-rickshaw graph.
- Reading the height of an x-t graph as the speed.The height is the position. The speed is the steepness of the slope.
- Taking a downward bend to mean slowing down.The bend gives the sign of the acceleration only. Compare the signs of \(v\) and \(a\) to decide speeding up or slowing down.
- Taking a curved x-t graph as a curved path.The horizontal axis is time. A body on a straight road can have any curved x-t graph.