You are seated in a train at a station. Through the window you see another train on the next track. Slowly, the other train begins to slide backwards. Then the platform slides backwards too, and you understand that your own train has started. For a moment you could not tell which train was moving. The passenger in the other train had the same doubt about you.
What kinematics is
Mechanics is the study of motion and of the forces that cause motion. Mechanics has two parts. Kinematics describes motion. Kinematics uses three quantities: position, velocity and acceleration. Kinematics does not ask which force produced the motion. Dynamics is the second part. Dynamics connects the motion to the forces. This chapter is kinematics only. Forces will come in a later chapter.
Position, origin and the observer
To describe motion, we first need the position of the body. Position must be measured from some fixed point. This fixed point is called the origin. The position of a body is the distance and direction of the body from the origin. On a straight line, one number is enough. The sign of the number gives the direction. In Fig 1.1 the particle is at \(x = +2.5\,\text{m}\). The plus sign means that the particle is on the positive side of the origin.
The person who measures the position is called the observer. The observer needs two things. The observer needs a coordinate system to measure position. The observer needs a clock to measure time. The coordinate system and the clock together are called a frame of reference.
A frame of reference is a coordinate system attached to an observer, together with a clock. Every position, every velocity and every acceleration is measured relative to some frame of reference.
A frame of reference is the answer to the question "measured from where, and by whose clock?". A student on a platform, a passenger in a train, and a pilot in an aircraft are three different observers. Each observer carries a different frame of reference.
Rest and motion
A body is at rest if the position of the body does not change with time. A body is in motion if the position of the body changes with time. Both statements need a frame of reference. The position is always measured in some frame. So rest and motion are always judged in some frame.
Look at the two readings in Fig 1.2. The first reading is the position of Priya measured from Ravi on the platform. This reading increases with time. In the frame of the platform, Priya is in motion. The second reading is the position of Priya measured from the door of the train. This reading stays at 1.5 m. It does not change. In the frame of the train, Priya is at rest. Both observers are correct. They are using different frames of reference.
No body is at absolute rest, and no body is in absolute motion. Rest and motion are properties of a body together with a frame of reference, never of the body alone. If a question does not name the frame, we take a frame attached to the Earth, for simplicity. A frame of reference is still there in every description of motion.
Students often say "the Earth is at rest". The Earth is at rest in a frame attached to the Earth. In a frame attached to the Sun, the Earth moves at about \(30\,\text{km/s}\). The sentence "the Earth is at rest" is incomplete until the frame is named.
A person sits in a bus that is moving along a straight road. Which statement is correct?
The map application on a phone reports a speed of \(0\,\text{km/h}\) when you stand still on the road. The application measures position relative to the Earth. If the same phone is used inside a moving train, the application reports the speed of the train, because the Earth frame is still being used. The application never reports the speed relative to the seat, although in that frame the passenger is at rest.
The point particle
A real body has a size. A car is about \(4\,\text{m}\) long. When the car travels \(100\,\text{km}\) from Jaipur to Ajmer, the length of the car does not matter for describing the journey. In such a case we treat the car as a single point that has mass but no size. This model is called a point particle.
The model does not always work. Consider a train of length \(600\,\text{m}\) crossing a bridge of length \(400\,\text{m}\). The length of the train is comparable to the length of the bridge. The time to cross the bridge depends on the length of the train. In this case the train cannot be treated as a point particle.
A train of length \(600\,\text{m}\) moves at a steady \(20\,\text{m/s}\). How long does the train take to cross a bridge of length \(400\,\text{m}\) completely? Compare with the answer obtained by treating the train as a point.
The train is completely on the bridge only after the last coach has entered, and has crossed only when the last coach has left.
So the front of the train must travel the length of the bridge plus the length of the train.
The distance to be covered is \(400 + 600 = 1000\,\text{m}\).
At \(20\,\text{m/s}\) the time is \(1000/20 = 50\,\text{s}\).
If the train were treated as a point, the distance would be only \(400\,\text{m}\), and the time would be \(20\,\text{s}\).
The point model gives a wrong answer here because the length of the train is not small compared with the bridge.
In which of these situations may the body be treated as a point particle?
A friend says, "The bus stop is at rest, so the bus stop is not moving at all." What is missing from this sentence?
The sentence does not say in which frame of reference the bus stop is at rest. In a frame attached to the Earth, the position of the bus stop does not change, so the bus stop is at rest. In a frame attached to a passing bus, the position of the bus stop changes every second, so the bus stop is in motion. Rest and motion always need a frame.
- Saying "at rest" or "in motion" without naming the frame of reference.Always add "relative to the ground", "relative to the train", or whatever frame is meant.
- Treating a long body as a point in a problem about crossing or overtaking.When the length of the body is comparable to the distances in the problem, include the length in the distance to be covered.