Motion in a PlaneIntroduction
Every morning Aman's uncle ferries pilgrims across the Ganga, from a small ghat on one bank to the ghat exactly opposite. Aman has watched him for years and noticed something odd: his uncle never points the boat at the ghat he wants to reach. He always aims a little upstream.
In the last chapter everything moved along a straight line, and a + or − sign was enough to say which way. A ferry on a flowing river is different: the engine pushes the boat one way, the current carries it another, and the boat ends up going a third way. A sign cannot describe that. Steer the ferry yourself. The engine drives the boat at 4 m/s in whatever direction it points, and the river flows at 2 m/s to the right. Drag the blue arrow to aim, then cast off.
The boat's real velocity was neither the blue arrow nor the green arrows, but a combination of the two that sent it along the purple track. To combine quantities that have a direction, we need a new kind of quantity with its own rules of addition: the vector. That is where this chapter begins.
From a Line to a Plane
First we learn the language of vectors: what they are, how to multiply them by numbers, how to add and subtract them by drawing and by calculation, and how to split one vector into parts along two directions. Then we use vectors to describe position, velocity and acceleration in a plane, and see that motion with constant acceleration is just two straight-line motions running side by side. That leads to two motions you see every day: projectile motion (a cricket ball, a jet of water) and uniform circular motion (a giant wheel, a ceiling fan).
Along a straight line, how many directions of motion are possible?
That is why signs are no longer enough. Everything in this chapter carries over to three dimensions, but we will mostly stay in a plane, where the pictures are easier to draw.