You sit in a train at New Delhi station. The train on the next track starts to move, and for a moment you feel that your own train has started. Later, both trains run side by side at the same speed. Through the window, the other train seems to stand still. The passengers in it look at you, and they do not move past your window.
A frame of reference, again
In the first chapter, a position was always measured from an origin, along an axis, with a clock. That set of origin, axis and clock is a frame of reference. A frame is usually fixed to some body: the platform, the train, the road. The person who measures from that frame is the observer.
In this chapter the frame fixed to the Earth is called the ground frame. The platform, the road and the railway track are all at rest in the ground frame. Most velocities you have met so far were measured in the ground frame, without saying so.
Priya walks in the train
Priya sits in a train that moves along a straight platform at \(10\,\text{m/s}\). Priya gets up and walks towards the front of the train. Measured along the floor of the train, Priya walks at \(1.5\,\text{m/s}\). Ravi stands still on the platform and watches.
Priya's train moves at \(10\,\text{m/s}\) past the platform, and Priya walks forward in it at \(1.5\,\text{m/s}\). Take the direction of the train as positive. In the frame of the train, what is the velocity of Ravi, who stands on the platform?
Play Fig 1.1 in Ravi's frame first. In \(4\,\text{s}\) the train moves \(40\,\text{m}\) along the platform. In the same \(4\,\text{s}\), Priya walks \(6\,\text{m}\) along the train. So Ravi sees Priya move \(46\,\text{m}\) in \(4\,\text{s}\), a velocity of \(11.5\,\text{m/s}\).
Now choose the train's frame and play again. The train stands still. Priya moves forward slowly, at \(1.5\,\text{m/s}\). Ravi and the platform move backwards at \(10\,\text{m/s}\), and Ravi soon leaves the picture. Nothing about the walk has changed. Only the frame has changed, and every velocity in the picture has changed with it.
The velocity of a body has a value only in a stated frame of reference. The same motion, measured in two frames that move relative to each other, has two different velocities. Both values are correct.
A velocity written without a frame is incomplete. When a problem says "a car moves at \(20\,\text{m/s}\)", it means \(20\,\text{m/s}\) in the ground frame. When a problem says "a man walks at \(1\,\text{m/s}\) on a moving walkway", the \(1\,\text{m/s}\) is measured in the frame of the walkway. Read which frame each number belongs to before you add anything.
The train moves forward at \(10\,\text{m/s}\). Priya now walks towards the back of the train at \(1.5\,\text{m/s}\), measured along the floor. What is the velocity of Priya in the platform frame, with the direction of the train positive?
Galileo asked his readers to shut themselves in the cabin below the deck of a large ship. Take flies, butterflies, a bowl of fish and a bottle that drips into a pan. Watch them while the ship is at rest. Then let the ship move at a steady speed. The flies still fly equally to every side. The drops still fall straight into the pan. No experiment inside the cabin can tell whether the ship moves or stands still. Galileo used this to answer people who said that a moving Earth would leave falling stones behind. The cabin is a frame of reference, and the laws of motion are the same in every frame that moves steadily.
The position of one body measured from another
Two bodies A and B move on the same straight line. Their positions in the ground frame are \(x_A\) and \(x_B\). Now put the origin on B itself. The position of A measured from B is the relative position of A with respect to B.
\[ x_{AB} = x_A - x_B \]
Read \(x_{AB}\) as "the position of A relative to B". A positive \(x_{AB}\) means A is on the positive side of B. A negative \(x_{AB}\) means A is on the negative side.
In Fig 1.2, A is at \(7\,\text{m}\) and B is at \(3\,\text{m}\). So \(x_{AB} = 7 - 3 = +4\,\text{m}\): A is \(4\,\text{m}\) ahead of B. With the second chip, A is at \(2\,\text{m}\) and B at \(8\,\text{m}\). Then \(x_{AB} = 2 - 8 = -6\,\text{m}\): A is \(6\,\text{m}\) behind B. The position of B relative to A is always the negative of this, \(x_{BA} = -x_{AB}\).
The origin does not matter
Suppose you move the origin of the ground frame \(5\,\text{m}\) to the left. Every position grows by \(5\,\text{m}\). Both \(x_A\) and \(x_B\) grow by the same amount, so their difference stays the same. The distance from B to A is a fact about the two bodies, not about the origin.
Think of two friends standing in a queue. You may count places from the front of the queue or from the ticket window across the road. The numbers you write for each friend change. The number of places between the two friends does not.
At one instant, a scooter A is at \(x_A = -2\,\text{m}\) and a bus B is at \(x_B = 5\,\text{m}\) on a straight road. Find the position of the scooter relative to the bus, \(x_{AB}\), in metres.
The position of a car relative to a truck is \(+30\,\text{m}\). What is the position of the truck relative to the car?
Two trains run side by side on parallel tracks, both at \(20\,\text{m/s}\) in the same direction. Explain why a passenger in one train sees the other train standing still.
The passenger measures in the frame of the passengers own train, and in that frame the own train is at rest. The other train moves forward in the ground frame exactly as fast as the own train. So the position of the other train, measured from the own train, does not change. A position that does not change with time means rest, so in this frame the other train is at rest.
- Writing a velocity without saying which frame it is measured in.Attach a frame to every number. "Priya, \(1.5\,\text{m/s}\) in the train frame" and "Priya, \(11.5\,\text{m/s}\) in the ground frame" are both correct.
- Writing \(x_{AB} = x_B - x_A\).The first subscript is the body, the second is the origin. Position of A relative to B is \(x_A - x_B\).
- Thinking that a body at rest on the ground is at rest in every frame.In the frame of a moving train, the platform and everyone on it move backwards.