Describing Motion Around Us: Class 9 Science · Chapter 4

Look out of a bus window for a minute and count what is moving. You might spot a scooter weaving between lanes, a ceiling fan in a shop, a child on a swing, crows circling a water tank and monsoon clouds sliding over the rooftops. Motion is everywhere, and most of it is complicated.

How can anyone describe all of that? The trick is to begin with simple, idealised motions and build up from them. You met three of them in earlier classes. They are motion along a straight line (linear motion), motion round a circle (circular motion) and to-and-fro motion about a fixed point (oscillatory motion). This chapter takes the first two much further. You will meet new quantities, displacement, velocity and acceleration, and learn to describe motion with numbers, graphs and equations as well as words.

4.1 Motion in a Straight Line

The simplest motion of all is along a straight line. A metro train on a straight stretch of elevated track moves along a straight line. So do a key slipping from your fingers to the floor and swimmers racing down their lanes. Before we can say how such an object moves, we need a way to say where it is at each moment.

4.1.1 Describing position

Suppose someone tells you the fruit cart is "forty metres away". Forty metres from where, and which way? To pin down a position we first choose a fixed point to measure from, called the reference point or origin. The position of an object is then given by two things: how far it is from the reference point, and in which direction.

Along a straight road there are only two directions, so we can draw the road as a number line with the origin O at zero. One direction is called positive and the other negative. By habit, a position on the right of O gets a + sign and one on its left a − sign. In Fig. 4.1 the bus stop is the origin: the cart is at +40 m and the dog at −40 m. Both are 40 m away, but the signs show they are on opposite sides.

Now we can say exactly what motion is. An object whose position, measured from the reference point, keeps changing as time passes is in motion. One whose position stays fixed is at rest. In the last step of Fig. 4.1, a few seconds later, the cart is still at +40 m, so it is at rest. The dog has gone from −40 m to −20 m, so it is moving. Whether something is at rest depends on the reference point you choose. A passenger asleep on a train is at rest compared with the train, but moving compared with the platform.

4.1.2 Distance travelled and displacement

Ananya cycles along a straight road from her home O to her friend's house F, 600 m to the east. Then she remembers she needs a notebook and rides 250 m back towards home, to a shop S (Fig. 4.2). How far has she travelled, and where has she ended up?

Add up every metre her wheels rolled: 600 m + 250 m = 850 m. This is the total distance travelled. It counts the whole path, whatever the direction. Now compare only where she started and where she stopped. The shop is 350 m east of her home. This net change of position is her displacement (विस्थापन): +350 m, taking east as positive.

To state a displacement fully you need its magnitude (the number with its unit, here 350 m) and its direction (east, or the + sign). A distance travelled needs only its magnitude. Both are measured in metres (m), the SI unit of length. On a trip that turns back, the distance keeps growing while the displacement can shrink. If Ananya rode on home, her displacement would fall to zero, although she had covered 1200 m.

4.1.3 Average speed and average velocity

Distance and displacement tell us how far an object went, but not how quickly. For that we divide by the time taken. You met average speed in Class 7: you find it by dividing the whole distance covered by the time taken to cover it.

Distance has no direction, so average speed has none either: it is a number with a unit and nothing more.

Some motions are steady and some are not. Look at the two walkers in Table 4.2. Walker 1 covers exactly 12 m in every 10 s, and if you timed shorter stretches you would find the same steady pattern. Motion like this, where every equal time interval brings an equal distance, is called uniform motion: the speed never changes. Walker 2 covers a little more in each interval than in the one before, so her motion is non-uniform. Other non-uniform motions slow down, or take turns at speeding up and slowing down.

Over the full 40 s, though, both walkers cover 48 m, so their average speeds are the same, $1.2\mathrm{m}{\mathrm{s}}^{-1}$. An average can hide a lot of detail.

Speed has one weakness: it ignores direction. Two autos both doing $30\mathrm{k}\mathrm{m}{\mathrm{h}}^{-1}$ may be driving towards each other or apart, and their speeds alone cannot tell you which. When direction matters we use average velocity (औसत वेग): the displacement divided by the time interval.

On a straight line the sign of the displacement carries over to the average velocity: a + displacement gives a + velocity, a − displacement a − velocity. Both averages are a length divided by a time, so both are measured in metres per second, written $\mathrm{m}{\mathrm{s}}^{-1}$ or m/s. Speedometers and road signs use kilometres per hour, $\mathrm{k}\mathrm{m}{\mathrm{h}}^{-1}$.

Average velocity is our first example of a rate of change: how much one quantity changes for each unit change in another. Here it is the change in position per second, so average velocity tells us how quickly, on average, the position changes with time.

4.1.4 Average acceleration

Ride a metro with your eyes shut and you can still tell when the train starts or stops. Yet at a steady speed on a straight track you could believe it was standing still. What you sense is not the velocity itself but a change in it. To measure how quickly velocity changes we use average acceleration.

If the velocity is $u$ at an earlier instant $t_1$ and $v$ at a later instant $t_2$, this becomes

The SI unit of acceleration is metre per second per second, written $\mathrm{m}{\mathrm{s}}^{-2}$ or m/s². An acceleration of $2\mathrm{m}{\mathrm{s}}^{-2}$ means the velocity changes by $2\mathrm{m}{\mathrm{s}}^{-1}$ every second.

Like displacement and velocity, acceleration has a direction. Along a straight line, when an object is speeding up, its acceleration points the same way as its velocity. When it is slowing down, its acceleration points the opposite way and so has the opposite sign (Fig. 4.4). In everyday speech slowing down is called deceleration; in physics it is simply an acceleration against the motion.

A velocity can change in size, in direction, or both, and each of these is an acceleration. In Section 4.4.1 you will meet a motion in which the speed never changes, only the direction, and yet there is an acceleration.

Here is a surprise: the fastest thing on the road can have the smallest acceleration. An express train cruising at a steady $120\mathrm{k}\mathrm{m}{\mathrm{h}}^{-1}$ on a straight track has zero acceleration, while a scooter crawling away from a signal may have quite a large one. Acceleration measures how quickly the velocity is changing, not how large it is.

In this chapter we deal only with motion at constant acceleration: the velocity changes by the same amount in every equal stretch of time, however short. Example 4.4 shows a motion of exactly this kind. Just as velocity can be given at an instant, so can acceleration; you will meet instantaneous acceleration in higher classes.

This constant acceleration of a freely falling object is the acceleration due to gravity, written $g$. Near the Earth's surface $g\approx 9.8\mathrm{m}{\mathrm{s}}^{-2}$, and a real falling object has this acceleration as long as air resistance is small. Taking the release point as origin and "down" as positive was our own decision; it simply kept every number positive. Any origin and either direction would work, as long as you stick with the same choice from the first line of a problem to the last. (In the yo-yo activity we also took downwards as positive; on a road we usually take the direction of travel.)

4.2 Graphical Representation of Motion

A table of readings describes a motion exactly, but it is hard to take in at a glance. A graph turns the same readings into a picture, and the eye is very good at pictures: a bend, a flat stretch or a steep climb jumps out at once. Later in this section you will also measure velocities and distances straight off a graph. You have drawn line graphs in mathematics; here we plot a quantity of motion, position or velocity, against time.

4.2.1 Plotting a graph

Let us plot the motion of a cyclist on a straight road, using the readings in Table 4.5.

4.2.2 Position–time graphs

Once a position–time graph is drawn, its shape tells the story of the motion. Fig. 4.7 shows the three basic shapes.

• A horizontal line means the position does not change, so the object is at rest. Note that it rests 25 m from the origin, not at the origin.
• A straight sloping line means equal displacements in equal times, so the velocity is constant.
• A curve that gets steeper means the displacement in each second keeps growing, so the velocity is increasing: the motion is accelerated.

A position–time graph tells you more than where the object is at each instant. It also tells you how fast the position is changing. The next activity shows how.

So the slope of a position–time graph is the velocity. A steeper line means a larger velocity, and a horizontal line has zero slope and means rest. For a curved graph the slope changes from point to point. Finding the velocity at a single instant from a curve needs a tangent line, which you will learn about in higher classes.

4.2.3 Velocity–time graphs

A velocity–time graph plots velocity up the y-axis against time along the x-axis, drawn in the same way as in Activity 4.3. Table 4.6 gives the velocities of three cars, each on a straight road.

Car A's graph is a horizontal line: its velocity never changes, so its acceleration is zero. Car B's graph is a straight line rising from zero. Its velocity grows by the same $4\mathrm{m}{\mathrm{s}}^{-1}$ in every 5 s: a constant acceleration in the direction of motion. Car C's line falls by $4\mathrm{m}{\mathrm{s}}^{-1}$ every 5 s: a constant acceleration against the motion, until the car stops.

Slope of a velocity–time graph

Apply the rise-over-run idea of Activity 4.4 to this new kind of graph. The rise is now a change of velocity and the run a change of time, so their ratio is the acceleration. For car B, between 5 s and 15 s the velocity rises from $4\mathrm{m}{\mathrm{s}}^{-1}$ to $12\mathrm{m}{\mathrm{s}}^{-1}$:

The same steps for car C give $a=-0.8\mathrm{m}{\mathrm{s}}^{-2}$, where the minus sign says the acceleration points against the velocity. Car A's line has zero slope, so its acceleration is zero.

Area under a velocity–time graph

Now look at the area between the graph line and the time axis. For car A between 0 s and 5 s, this region is a rectangle (Fig. 4.12). Its height is the velocity, $12\mathrm{m}{\mathrm{s}}^{-1}$, and its width is the time, 5 s. Their product, 60 m, is exactly how far car A travels in those 5 seconds.

The same holds for every velocity–time graph, whatever its shape: over any stretch of time, the area trapped between the graph and the time axis is the displacement. For car B from 5 s to 15 s the region is not a rectangle, but you can split it into two pieces. The rectangle at the bottom is 10 s wide and $4\mathrm{m}{\mathrm{s}}^{-1}$ tall, which gives 40 m. The triangle on top has a base of 10 s and a height of $12-4=8\mathrm{m}{\mathrm{s}}^{-1}$, so its area is $\frac{1}{2}\times 10\times 8=40\mathrm{m}$. Car B's displacement between 5 s and 15 s is 40 m + 40 m = 80 m.

So one velocity–time graph answers two questions: how sharply the velocity changes (read the slope) and how far the object goes (measure the area). Try both in the motion lab below.

4.3 Kinematic Equations for Motion in a Straight Line with Constant Acceleration

Graphs are good for seeing a motion; equations are better for calculating it. Constant acceleration has a handy consequence: there is no difference between the acceleration at an instant and its average over a stretch of time, so (4.3c) holds over any interval we like. That lets us link five quantities: the initial velocity $u$, the final velocity $v$, the acceleration $a$, the time interval $t$ and the displacement $s$. We start the clock, $t=0$, at the instant the velocity is $u$.

Velocity after a displacement s. Sometimes the time is not known, so we want an equation without $t$. Square both sides of (4.4a):

Take $2a$ out of the last two terms: $2uat+a^2{t}^2=2a(ut+\frac{1}{2}a{t}^2),$ and the bracket is exactly $s$ by (4.4b). So

These three are the kinematic equations, also called the equations of motion. Give them any three of the five quantities and they hand back the other two: where a train will be after a minute, or how fast a falling coconut is going when it lands.

4.4 Motion in a Plane

So far everything has moved along one line: motion in one dimension. Many motions need two directions to describe them. Think of a paper plane gliding across a classroom, a carrom striker bouncing off the edge of the board, or the tip of a ceiling-fan blade. These are motions in a plane, or in two dimensions.

4.4.1 Uniform circular motion

When an object moves along a circular path, its motion is circular. Ishaan cycles anticlockwise round a circular track in a park; the radius of the track is 35 m (Fig. 4.16).

Going a quarter of the way round, from A to B, Ishaan covers a quarter of the circumference, 55 m (taking $\pi =22\mathrm{/}7$). His displacement is the straight chord AB, only about 49.5 m. At C, half-way round, he has covered 110 m and is one diameter, 70 m, from A. After a full round he has covered the whole circumference, $2\pi R=220\mathrm{m}$, yet his displacement is zero.

If one round takes a time $T$, his average speed over a round is the circumference divided by $T$:

His average velocity over a full round is zero, because his displacement over a full round is zero.

Now suppose Ishaan holds his speed perfectly steady. Motion round a circle at constant speed is called uniform circular motion. Its speed never changes. Does its velocity stay the same too?

Compare driving round a square city block with driving round a circular road (Fig. 4.17). On the block the car goes straight along each side and changes direction only at the four corners. A block with more, shorter sides would need more turns, each gentler. A circular road is the limit of this: it has no straight piece at all, so the direction of the velocity changes at every instant, even if the speedometer needle never moves.

Draw a line that just grazes the circle at the release point, touching it there and nowhere else. This is the tangent at that point, and it is exactly the marble's escape route. The same is true at every point of a circular path: the velocity always lies along the tangent, pointing the way the object is moving.

So is a car taking a curve at a steady $40\mathrm{k}\mathrm{m}{\mathrm{h}}^{-1}$ accelerating? Its speedometer says nothing is changing. But velocity includes direction, and the direction is changing every moment; a changing velocity is, by definition, an acceleration (Section 4.1.4). Uniform circular motion is therefore accelerated motion, even though the speed never budges. You can feel it, too: passengers in a car on a sharp curve are pressed towards the outside of the bend.

Hardly any real path is a perfect circle travelled at a perfectly steady speed: the Earth's orbit is very slightly oval, and no driver holds the needle exactly still. Physicists still use uniform circular motion as a first model for satellites, the Moon, the planets and bends in a road, and add the corrections later.