Activity: Galilean relativity

Aims

  • Understand how an event can be measured in different frames of reference.
  • Understand the difference between inertial and non inertial frames of reference.
  • Apply the Galilean transformations.
  • Understand how the nature of light poses problems for Galilean relativity

Frames of Reference

The position and time for an event can be measure by an observer in a frame of reference. This is a coordinate system (for measuring position) and clocks (for measuring time). Its is important to have a lot of clocks so that the time of an event can be recorded at the place where it happens.

Let us consider two observers Alice A and Alice B.

B is moving with velocity v relative to A as shown.
Each is going to measure the time and place of an event happening in front of them. The event is the ball falling to the ground
Each lives in a world with only 1 dimension of space (x) and time (t)
Each has her own frame of reference to measure the event, this is a ruler and set of clocks.

This is A's frame of reference

B's is the same but moving to the right.

At time t = 0 s A and B are in the same place.
Let's watch them both observer the event.

The animation was stopped at the time of the event.
The ruler is marked in meters and one revolution of the clock is 60 s.

  • What is the velocity of A relative to B?
  • At what time does the event take place according to A?
  • At what time does the event take place according to B?
  • At what position does the event take place according to A?
  • At what position does the event take place according to B?

Galilean Transformations

Position and time transformation

In general an event measured by A occurs at position x and time t. The same event measured by B occurs at position x' and time t'.

  • Write an expression relating x and x'.

An equation that converts a measurement in one frame of reference to another is called a transformation.

  • Is there any difference between the t and t'?

Velocity transformation

Consider a ball travelling along the x axis as shown.

At t = 0 the displacement of the ball is zero as measured by both A and B

  • What is the displacement at the end of the clip as measured by A?
  • What is the displacement as measured by B?
  • What is the time taken measured by A?
  • What is the time taken measured by B?
  • What is the velocity measured by A?
  • What is the velocity measured by B?
  • If the velocity measured by A is u write an equation for the velocity u' measured by B.

    u' = u - v

Acceleration transformation

Observer A observes a body accelerating from rest to velocity u in t seconds.

  • What is the acceleration measured by A?
  • What is the initial velocity measured by B?
  • What is the final velocity measured by B in terms of u?
  • What is the acceleration as measured by B?

The acceleration is the same for both a = u/t

Newtons second Law

The acceleration, a in the previous example was caused by a force F.

  • If the mass of the ball is m apply Newtons second law to derive an equation relating F and a.
  • Is this equation the same for both A and B?

Newton's second law and all other laws of physics are the same for all inertial frames of reference.

Non Inertial frames of reference.

Consider A standing in a train accelerating away from a station as B watches.

Notice the ball on the ground.

  • Are there any forces acting horizontally on the ball?
  • Does the ball accelerate according to B?
  • Does the ball accelerate according to A?

Let's have a look at that from A's frame of reference.

  • According to A is Newton's second law obeyed?

The nature of light

In 1864 James Clark Maxwell deduced that light is an electromagnetic wave with velocity 3 x 108 ms-1 with no requirement of any medium. Further experiment verified that the velocity of light was indeed the same for all inertial observers independent of relative motion, this does not agree with the Galilean transformations. Consider the Alice's measuring the velocity of light.

According to Galilean transformations.

  • What is the velocity of light measured by A?
  • What is the velocity of light measured by B?
  • Why doesn't this agree with Maxwell?

There is also a fundamental problem with electromagnetism.

When current flows through a wire it produces a magnetic field, in fact all moving charge produce magnetism.

An electron moves as shown

  • According to A is the electron moving?
  • According to A does the electron produce a magnetic field?
  • According to B is the electron moving?
  • According to B does the electron produce a magnetic field?

It appears that the Galilean transformations don't work.

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