0625 · 1.2Motion

Distance, speed and velocity

0:00A cyclist can travel a long distance slowly, or a short distance quickly. The distance alone does not describe the motion. We also need the time taken, and sometimes the direction of travel. In this video, we will connect those ideas to graphs, calculations and falling objects. We need just a few starting ideas. Distance measures how far an object travels, usually in metres. Time measures how long the motion takes, usually in seconds.

0:24On our graphs, time goes on the horizontal axis. Reading the vertical axis carefully will tell us whether we are looking at distance or speed. Speed is distance travelled per unit time. The phrase per unit time means how much distance is travelled during each second, or another chosen unit of time. Speed equals distance divided by time. Dividing spreads the distance across the time taken, giving the distance travelled for each unit of time.

0:50Suppose a cyclist travels one hundred and fifty metres in thirty seconds. We calculate speed by dividing one hundred and fifty by thirty. The result is five metres per second. This means the cyclist travels five metres each second if the speed is constant. Always use compatible units before calculating. A distance in metres divided by a time in seconds gives metres per second. If time is given in minutes, convert it into seconds first when metres per second is required.

1:16One minute contains sixty seconds, so two minutes becomes one hundred and twenty seconds. Velocity is speed in a given direction. Two cyclists travelling at five metres per second have the same speed. If one travels east and the other west, their velocities are different. Direction is part of velocity, so changing direction changes velocity even when the speed stays the same.

Calculating average speed

1:35A journey rarely happens at one unchanging speed. You might cycle quickly, slow down, and then stop at a crossing. Average speed equals total distance travelled divided by total time taken. It describes the whole journey using one value, rather than telling us the speed at every moment. Imagine travelling one hundred and twenty metres in twenty seconds, then returning the same distance in forty seconds. You also stop for twenty seconds during the journey.

2:00The total distance is two hundred and forty metres. The total time is eighty seconds, including the stop, so the average speed is three metres per second. Returning to the starting point does not make the distance travelled zero. Both parts of the journey count because distance records how far you actually travel. You also cannot usually find average speed by averaging two speed values. Different stages may last different times, so total distance divided by total time is the reliable method.

Distance-time graphs

2:25A distance time graph shows the distance travelled as time passes. A straight rising line means equal distances are covered in equal times. That represents constant speed. A horizontal line means time passes without any additional distance being travelled, so the object is at rest. The steepness of the line tells us the speed. A steeper line represents more distance travelled during the same time interval. The mathematical name for steepness is gradient.

2:50On a distance time graph, gradient equals speed because it compares a change in distance with the corresponding change in time. Let us calculate the speed along the straight rising section shown. Choose two well separated points on that section, at four seconds and eight seconds. The distance increases from twelve metres to twenty four metres, giving a change of twelve metres. The time increases by four seconds, so the speed is 12 ÷ 4, or three metres per second.

3:16Notice that we used changes between two points, rather than simply dividing their coordinates. This method works even when a straight section does not begin at the origin. Draw a large gradient triangle to make those changes clear. Its vertical side represents the distance change, and its horizontal side represents the time change. A curve means the gradient is changing, so the speed is changing. When a distance time graph becomes steeper, the object is accelerating because its speed increases.

3:42When it becomes flatter, the object is decelerating because its speed decreases. Judge the changing steepness, not simply whether the line is high or low. Data can reveal the same patterns without a graph. Equal distances travelled in equal time intervals indicate constant speed. Increasing distances in successive equal intervals indicate increasing speed. Decreasing distances indicate slowing down, while no increase in total distance indicates rest. To plot a graph, label both axes with the quantity and unit.

4:13Choose a clear scale that uses a substantial part of the available grid. Plot each pair of values carefully, then draw the line or curve supported by the motion. For a sketch, the correct shape matters more than exact numerical positions.

Speed-time graphs and acceleration

4:26A speed time graph tells a different story because its vertical axis shows speed directly. A horizontal line above zero means constant speed, not rest. A line along the time axis means zero speed, so the object is stationary. An upward sloping section shows the object accelerating. A downward sloping section shows it decelerating. Keep the two graph types separate. On a distance time graph, height tells us the distance travelled so far.

4:52On a speed time graph, height tells us the speed at that moment. The same horizontal shape therefore has different meanings because the vertical quantities are different. We now move into Supplement content for Extended students. Acceleration is change in velocity per unit time. Acceleration equals change in velocity divided by time taken. For motion in one fixed direction, we can calculate this using the change in speed, since the direction remains unchanged.

5:19Suppose a car increases its velocity from two metres per second to ten metres per second in four seconds. First, subtract the initial velocity from the final velocity. 10 − 2 gives a change of eight metres per second. Divide eight by four to obtain an acceleration of two metres per second squared. Metres per second squared describes how quickly velocity changes. An acceleration of two metres per second squared means velocity increases by two metres per second during each second.

5:46With constant acceleration, equal time intervals produce equal changes in velocity. The speed increases steadily rather than remaining constant. On a speed time graph for motion in one direction, gradient equals acceleration. The vertical change gives the change in speed, and the horizontal change gives the elapsed time. Dividing those changes produces the acceleration. A straight sloping section therefore represents constant acceleration because its gradient stays the same. A curved speed time graph represents changing acceleration because its gradient changes.

6:17If a rising curve becomes less steep, speed is still increasing, but acceleration is decreasing. From data, equal velocity changes in equal time intervals indicate constant acceleration. Unequal changes over equal intervals indicate changing acceleration. Deceleration is a negative acceleration. For these straight line examples, take the forward direction as positive. A car slowing from twelve metres per second to four metres per second has a velocity change of negative eight metres per second.

6:45The negative sign comes from subtracting a larger initial velocity from a smaller final velocity. If that change takes four seconds, acceleration equals −8 ÷ 4. The answer is negative two metres per second squared. We can also describe the magnitude of its deceleration as two metres per second squared. When asked for acceleration, retain the negative sign to show the velocity is decreasing in our chosen positive direction.

Areas under speed-time graphs

7:10The area under a speed time graph gives the distance travelled. At constant speed, the area is a rectangle with speed as its height and time as its width. Rectangle area is height multiplied by width. This is exactly distance equals speed multiplied by time, so the graph expresses the same relationship. For a speed of six metres per second lasting five seconds, the rectangle has area 6 × 5.

7:33The distance travelled is thirty metres. The units also support this result because metres per second multiplied by seconds gives metres. Here, the area under the graph stands for a distance in metres. Now consider a complete journey with three stages. The object accelerates uniformly from rest to eight metres per second during four seconds. It then travels at that speed for six seconds. Finally, it slows uniformly to rest during two seconds.

7:58The first region is a triangle because speed increases steadily from zero. Triangle area equals half multiplied by base multiplied by height. Half multiplied by 4 × 8 gives sixteen metres. The triangle is half the corresponding rectangle because the object spends the interval building up to its final speed. The middle rectangle represents six seconds at eight metres per second. Its area is 6 × 8, giving forty eight metres.

8:24The final triangle has base two seconds and height eight metres per second. Half multiplied by 2 × 8 gives eight metres during the slowing stage. Add the three distances because each region represents part of the same journey. 16 + 48 + 8 gives seventy two metres. The total time is twelve seconds. Average speed is therefore 72 ÷ 12, giving six metres per second. This example also connects area and gradient without confusing them.

8:51The first section has gradient 8 ÷ 4, giving acceleration of two metres per second squared. Its area gives sixteen metres travelled, which is a different quantity. A gradient describes how speed changes, while an area adds up the distance covered.

Falling objects and terminal velocity

9:04We can now apply these ideas to falling objects. Near the surface of the Earth, the acceleration of free fall is approximately constant. Its value is approximately nine point eight metres per second squared. This is acceleration, not a fixed falling speed, so an object in free fall keeps gaining speed. The rest of this falling objects section is Supplement content for Extended students. The gravitational field is approximately uniform over a small region near the surface.

9:31This means its strength, and the acceleration it produces in free fall, are approximately unchanged throughout that region. Without air resistance, objects released together have the same downward acceleration regardless of mass. Their speeds increase by about nine point eight metres per second each second. In air, a falling object also experiences air resistance acting upwards. When released from rest, its speed and air resistance are initially zero. As it gains speed, air resistance increases, so its downward acceleration becomes smaller.

10:00Eventually, air resistance can become equal to the downward gravitational force. The forces then balance, so acceleration becomes zero. The object then falls at a constant velocity. This is called terminal velocity. Terminal velocity does not mean the object stops, because zero acceleration means no change in velocity rather than zero velocity. A similar pattern can occur when an object sinks through a liquid. Liquid resistance opposes its motion and increases as its speed increases.

10:29The liquid also provides an upward force called upthrust. Terminal velocity is reached when the total upward forces balance the downward gravitational force. Watch for three especially common mistakes. A falling object approaching terminal velocity can still gain speed while its acceleration decreases. A speed time graph flattening above zero therefore does not show the object stopping. Also, free fall acceleration is approximately constant, but resistance can make the actual acceleration smaller than that value.

10:58An exam question might ask you to describe this motion, calculate acceleration, and determine distance travelled. Start with the graph shape before calculating. The object decelerates at a constant rate because the graph is a straight line with a negative gradient. It reaches rest after three seconds because its speed becomes zero. The change in velocity is 0 − 12, giving negative twelve metres per second. Divide this by three seconds to obtain negative four metres per second squared.

11:25The distance travelled is the triangular area beneath the graph. Half multiplied by 3 × 12 gives eighteen metres. For a terminal velocity explanation, give each physical step clearly. As the object speeds up, air resistance increases. The resultant downward force decreases, so acceleration decreases. When air resistance equals the gravitational force, acceleration is zero and the object falls at constant terminal velocity. The central idea is that motion describes both how far an object travels and how its velocity changes.

11:56Speed connects distance with time, while acceleration connects velocity change with time. Distance time gradients give speed. Speed time gradients give acceleration, and speed time areas give distance travelled. Read the axes before interpreting any graph, and keep constant speed separate from constant acceleration. Include every stage and every stop when calculating average speed. For falling objects, distinguish constant free fall acceleration from motion with resistance. These connections let you explain the motion rather than simply recognise a formula.