Motion in a Straight LineIntroduction
Seven in the morning. Aman cycles along the straight canal road to school with Sina riding on the carrier. A kite drifts overhead, the Metro glides past on its viaduct and a neem tree drops its leaves. Even when Aman stops at the crossing, air keeps moving in and out of his lungs and blood keeps flowing through his veins. The Earth under him spins once a day and goes round the Sun once a year, and the Sun itself moves through the Milky Way.
Motion is everywhere. Motion is a change of position with time, and this chapter is about describing it: how fast the position changes (velocity) and how fast that changes (acceleration). We keep to the simplest case, motion along a straight line, called rectilinear motion. When the acceleration is uniform, a short set of equations describes everything, and at the end we see that motion also depends on who is watching: relative velocity. Describing motion without asking what causes it is called kinematics. The causes come in Chapter 4.
One simplification makes all of this easier. We treat the moving body as a point object. That is fair whenever the body is much smaller than the distance it moves in the time we are interested in, and its turning or shape does not matter. Sort these: which bodies can be treated as points?
The train is half a kilometre long, but its trip is 1400 km. The Earth is huge, but its orbit is more than ten thousand times wider. The top, the gymnast and the door hardly travel at all: what they do is turn, and a point cannot turn. From now on every moving body in this chapter is a point.
Where Is Sina?
To say where a body is, we need a frame of reference: a fixed point called the origin, a line through it (our x-axis), a choice of which way along the line is positive, and a clock. Take a straight road with Sina's home, a chai stall 300 m along it, and a school 800 m from home. Sina is waiting at the chai stall. Her position x is her distance from the origin, with a + sign on the positive side of the origin and a − sign on the other side.
Same Sina, same spot. Choose the origin and the positive direction to give her position each of these values.
The origin and the positive direction are yours to choose, but you must choose them, and say so, before you write down a single position, displacement, velocity or acceleration. In this chapter the road is the x-axis, and unless a problem says otherwise the origin is where the motion starts and the positive direction is the way the body sets off.
Path Length and Displacement
"How far did you walk?" and "how far are you from where you started?" are different questions. The first asks for the path length: the total length of road covered, every step counted. It is never negative. The second asks for the displacement: the change in position,
which carries a sign and ignores everything that happened in between. Throughout this chapter, position and displacement are blue, velocity is green, acceleration is purple, and averages and path length are amber.
Walk Sina along the road, one second per move. Amber arcs count the path length, a blue arrow shows the displacement, and the graph plots her position x against the time t.
Look back at your walks. The amber arcs only ever add up, so the path length is never smaller than the length of the blue arrow. The two are equal only when Sina never turns back.
The blue graph is a position–time graph, or x–t graph. A flat piece means standing still. A rising piece means moving in the +x direction, a falling piece moving in the −x direction. Steady walking gives straight pieces: equal displacements in equal times, which is called uniform motion. The steeper a straight piece, the faster the motion, and its slope Δx/Δt is the velocity.
Speed Round: Read the x–t Graph
Every x–t graph tells a story. A flat line is a body at rest, a steeper line a faster body, and the slope Δx/Δt of a straight line is the velocity: up is positive, down is negative.
Get five right to move on.