Acceleration and velocity-time graphs

Picture two cars pulling away from the same set of lights, side by side. Ten seconds later, both are doing the same speed. Yet for the first five seconds, one was pulling away twice as hard as the other. A glance at either speedometer, at any moment, cannot tell you that. The number you want is not the speed. It is how fast the speed is changing.

This is video three of five in Motion graphs and acceleration. If velocity itself feels shaky, the video called Velocity comes first. You do not need Speed at an instant from a tangent to follow this one.

That number is the acceleration. Acceleration is the change in velocity divided by the time taken, and it is on Foundation and Higher papers alike. The two words often get blurred. Velocity is how fast you are going in a direction, in metres per second. Acceleration is how fast that velocity is changing, in metres per second squared. That unit sounds odd, so read it slowly: metres per second, per second. It counts how many metres per second you gain in each second. Its symbol is a. In symbols, a equals v minus u, all over t, where v is the final velocity, u is the initial velocity, and t is the time taken. You are expected to know this equation by heart. Rearranged, the change in velocity is acceleration times time, and the time is the change in velocity divided by the acceleration. Take a tram. It speeds up from four metres per second to twenty-two metres per second in twelve seconds. What is the tram's acceleration, in metres per second squared? The acceleration is one and a half metres per second squared. Write the equation first: a equals v minus u, over t. The change in velocity is twenty-two take away four, which is eighteen metres per second. Divide by twelve seconds, and you get one and a half metres per second squared. Now the same eighteen metres per second of change takes twenty-four seconds, instead of twelve. Does the acceleration double, halve, or stay the same? It halves, to three quarters of a metre per second squared. The time is on the bottom of the division, so twice the time means half the acceleration. It helps to know rough sizes. A car pulling away gains about two or three metres per second each second, and a lift starting up about one. Measuring one in the lab is the required practical on force and acceleration. When something slows down, its velocity falls, so the change in velocity is negative. A train slows from twenty-five metres per second to a stop in eight seconds. The change in velocity is zero take away twenty-five, which is minus twenty-five metres per second. Divided by eight seconds, that is minus three point one metres per second squared, to two significant figures. What is that minus sign telling you about the train? That it is slowing down. The acceleration points against the motion, so the train loses speed as it goes. If a question asks for the deceleration, which means the slowing-down rate, drop the sign, because the word already says it. The deceleration is three point one metres per second squared.

Acceleration has its own graph. A velocity-time graph has velocity up the side, in metres per second, and time along the bottom, in seconds. It looks a lot like a distance-time graph, which is why the two get mixed up. Here they are side by side, each with a flat stretch. What does the flat stretch mean on the distance-time graph, and on the velocity-time? On the distance-time graph, flat means stopped. On the velocity-time graph, flat means still moving, at a steady velocity. Flat on velocity-time means the velocity is not changing. It is not zero. It holds at the same value, so the object keeps going. One examiner's report on a Foundation paper puts it plainly. A common error was to see the horizontal part of the graph as being where the train was stationary, even though the first part had been correctly described in terms of increasing velocity. The fix is to read the axis label before the line. If it says velocity, a flat line means moving at a steady velocity. With the distance-time graph put away, here are the two sentences, one under the other. The gradient of a distance-time graph is the speed. The gradient of a velocity-time graph is the acceleration. Why is the gradient of a velocity-time graph the acceleration? Because its triangle's rise is a change in velocity and its run is a change in time. Change in velocity over time taken is what acceleration means. It is the same triangle on a new graph. On velocity-time it gives the acceleration, and a flat line means still moving. A report on a Higher paper records that many candidates did not realise that acceleration is the rate of change of velocity with time and is therefore the gradient of the graph. So the habit is one rule. If a question asks how fast something is changing, you want the slope. The same graph answers a different question in Distance from the area under a velocity-time graph.

Velocity-time lines come in four shapes. Sloping up means speeding up. Sloping down means slowing down. Flat means a steady velocity. And a straight sloping line means the acceleration is uniform, which means it stays the same every second, because the slope never changes. Here is a description that sounds right: the ball accelerates at a constant speed. It gets a cross. What is wrong with saying the ball accelerates at a constant speed? A constant speed would be a flat line, with no acceleration at all. The phrase describes two different graphs at once. The phrase that works is accelerates uniformly, or accelerates at a constant rate. Both name a slope that does not change. A second Foundation report concerns students who said a ball accelerated. It says very few of these recognised that a straight line velocity-time graph represents a uniform acceleration and so failed to score the second mark available. Comments such as the ball accelerates at a constant speed did not score a mark. So whenever the line is straight and sloping, write the word uniformly. It names the steady slope in one word.

Now find an acceleration straight off the graph. One straight section runs from ten seconds at six metres per second, up to thirty seconds at eighteen metres per second. The triangle and the division are the same as for the tram. The new step is that neither corner sits at zero, so both changes come first. Pause and find that section's acceleration, in metres per second squared. It comes out at nought point six metres per second squared. The change in velocity is eighteen take away six, which is twelve metres per second. The change in time is thirty take away ten, which is twenty seconds. Twelve metres per second divided by twenty seconds is nought point six metres per second squared. One last graph, in three sections. From zero to six seconds it climbs from rest to twelve metres per second. From six to sixteen seconds it stays flat. From sixteen to twenty seconds it falls back to zero. How would you describe each of the three sections, before any sums? First, it accelerates uniformly. Then it moves at a steady velocity, not stopped. Last, it decelerates uniformly until it stops. Now, judging by eye, which section has the biggest acceleration, ignoring the sign? The third one. It has the steepest slope, so it changes speed fastest of the three. The numbers agree. The climb is twelve over six, two metres per second squared. The fall is twelve over four, a deceleration of three metres per second squared. The flat part is zero. An examiner's report on a Higher paper has this line. Most candidates scored the mark in this question. Candidates should relate the gradient of the velocity time graph to the acceleration and make a comment in terms of the steepest gradient. So when you compare, name the steepest gradient, and say that it means the biggest acceleration.

Before you go, test yourself on three of these. The question comes first, the answer second. What does the gradient of a velocity-time graph give you? The acceleration. The gradient of a velocity-time graph is the acceleration. Back at the lights now. One car goes from rest to twenty metres per second in five seconds. What is that car's acceleration, in metres per second squared? Four metres per second squared: the whole change, divided by the time it took. Different case: a velocity-time line is flat at eight metres per second. Stopped, or moving? Moving, at a steady eight metres per second. Flat on velocity-time is still going, not stopped. Same triangle, new graph: on velocity-time, the gradient is the acceleration. And the two cars at the lights? The hard starter gained four metres per second each second, the other only two. No speedometer shows that, but the gradient does.

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Next in the chapter: Distance from the area under a velocity-time graph.

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