A5 notes: Reference frames
When two cars travelling at 15 m s-1 approach one another, one driver observes the approach of the other as being at 30 m s-1.
When two particles in an accelerator travelling at 2 x 108 m s-1 approach one another, based on the physics you have studied so far, you could reasonably infer that the relative speed would be 4 x 108 m s-1. But this is impossible.
Why? The speed of light places a speed limit on the universe: 3 x 108 m s-1
Reference frames
In physics, a reference frame (or frame of reference) is a perspective used to observe and measure physical phenomena. It consists of a coordinate system along with a clock to quantify positions and times. There are two main types of reference frames: inertial and non-inertial.
Inertial Reference Frames: These are frames in which an object remains at rest or moves at a constant velocity unless acted upon by an external force. Newton's first law of motion holds true in such frames. A good example of an inertial frame is a stationary or smoothly moving vehicle (ignoring air resistance and friction).
Non-Inertial Reference Frames: These frames are accelerating or rotating, so Newton’s laws do not apply directly without correction. In these frames, fictitious forces, like the Coriolis or centrifugal forces, are sometimes introduced by non-physicists to explain motion. For instance, inside a turning car, a person feels pushed to one side - an effect of observing from a non-inertial frame.
The choice of reference frame is important in describing and predicting the motion of objects. While many physical laws hold universally, the equations of motion can look different depending on the frame used. In classical mechanics, Newton’s laws are valid in inertial frames, while special adjustments must be made in non-inertial ones.
In advanced physics, particularly in Einstein’s theory of relativity, reference frames play a critical role. Special relativity, for example, deals with how observers in different inertial frames perceive time, space, and simultaneity differently, especially when moving at speeds close to the speed of light.
Ultimately, reference frames help define how we measure and understand motion, forces, and time, making them fundamental in many areas of physics.
Galilean relativity
Newton's laws of motion are the same in all inertial reference frames and this is known as Galilean relativity.
Position and time
In Galilean relativity, the transformation between two reference frames moving relative to each other at a constant velocity is described by simple relationships for space and time. These transformations show how the coordinates of an event in one frame relate to those in another.
Consider two reference frames:
- S: A stationary frame of reference.
- S′: A frame of reference moving at a constant velocity \(v\) along the \(x\)-axis relative to S.
The position of an object in the two frames is given by the following coordinates:
- In the stationary frame S, the position is \(x\).
- In the moving frame S′, the position is \(x′\).
To find the relationship between the coordinates in the two frames, we note that after a time \(t\), the moving frame S′ will have shifted by a distance \(vt\) relative to S (since the velocity of S′ is \(v\)).
Spatial Transformation:
The position \(x′\) in the moving frame S′ is obtained by subtracting the distance the frame S′ has traveled from the position \(x\) in the stationary frame S:
\(x′=x−vt\)
This equation shows that an object's position in the moving frame S′ is its position in the stationary frame S, minus the distance the moving frame has traveled over time \(t\). This makes sense because if the object is stationary in the frame S, in the moving frame S′ it appears to move backward by a distance \(vt\).
Temporal Transformation:
In Galilean relativity, time is assumed to be absolute. This means that the time interval \(t\) is the same in all reference frames, regardless of their relative motion. Therefore, the time \(t′\) in the moving frame is simply equal to the time \(t\) in the stationary frame:
\(t′=t\)
Summary of Galilean Transformation:
The Galilean transformation equations between the two reference frames are:
- \(x′=x−vt\) (for spatial transformation)
- \(t′=t\) (for time transformation)
These transformations are valid for low velocities compared to the speed of light. At higher velocities, the more complex Lorentz transformations from Einstein’s special relativity replace the Galilean transformations.
- \(x′\) is the position measured in the moving frame
- \(x\) is the position measured in the stationary frame
- \(v\) is the speed of the moving frame
- \(t\) is the time measured in the stationary frame
- \(t′\) is the time measured in the moving frame
Velocity addition
The Galilean transformation equations for position and time can be used to derive the velocity addition equation in classical mechanics. This equation describes how the velocity of an object as seen from a moving frame of reference relates to its velocity as seen from a stationary frame.
Derivation of the Velocity Addition Equation
Consider two reference frames:
- S: the stationary frame.
- S′: the frame moving with a constant velocity \(v\) relative to S along the \(x\)-axis.
Let’s suppose an object is moving with velocity \(u\) in the S frame. We want to find the velocity \(u′\) of the object as observed from the moving frame S′.
Step 1: Using the Galilean Transformation for Position
The Galilean transformation for position is:
\(x′=x−vt\)
where:
- \(x′\) is the position of the object in the moving frame S′
- \(x\) is the position of the object in the stationary frame S
- \(v\) is the velocity of the moving frame S′ relative to S
- \(t\) is the time, which is the same in both frames (since \(t′=t\))
Step 2: Taking the Time Derivative
To find the velocity, we differentiate both sides of the position equation with respect to time. The velocity is the rate of change of position with respect to time, so we have:
\(\frac{dx′}{dt} = \frac{dx}{dt} - v\)
In terms of velocities:
- u′=dx′dtu′ = \frac{dx′}{dt}u′=dtdx′ is the velocity of the object in the moving frame S′,
- u=dxdtu = \frac{dx}{dt}u=dtdx is the velocity of the object in the stationary frame S,
- v is the constant velocity of the moving frame.
Thus, the equation becomes:
\(u′=u−v\)
This is the velocity addition equation in Galilean relativity.
- \(u\) is the velocity of the object in the stationary frame S.
- \(u′\) is the velocity of the object in the moving frame S′.
- \(v\) is the relative velocity between the two frames.
The equation shows that the velocity of the object as measured in the moving frame S′ is the velocity of the object in the stationary frame S, minus the velocity of the frame S′ relative to S.
Example:
If you’re on a train moving at velocity \(v=10\) m s-1 and you observe a ball rolling forward inside the train at \(u′=2\) m s-1, the velocity of the ball relative to someone standing on the platform would be \(u=u′+v=2\) m s-1\(+10\) m s-1\(=12\) m s-1.