Unit 1 Mechanics

1.1 Kinematic relationships and relativistic motion

Learning Outcomes

  1. Derive from a = dv⁄dt ie a = d2s⁄dt2 the kinematic relationships: v= u +at, s = ut + ½at2 and v2 = u2 +2as.
  2. Carry out calculations involving constant accelerations.
  3. State that the greatest possible speed of a body is that of light in vacuo.
  4. State that the relativistic mass m of a moving body is not constant but increases with speed.
  5. Carry out calculations involving mass and speed, given the formula m = mo⁄ √(1 - v2⁄c2)
  6. State that the relativistic energy E of an object is mc2.

Aids to Understanding

Working with indefinfinite integrals

Basic knowledge of differentiation and integration is assumed. An indefinite integral is one with no upper or lower limits. In general the answer to an integration of this type always has a constant of integration. This is because an integral is actually an anti-derivative, thus when we differentiate it we end up where we started. The deivative of a constant is zero. If f(x) = x2 then its integral is 1/3x3 + constant. If we differentiate 1/3x3 + constant we are back to x2, ie where we started.

The equations of motion

This problem is about a rocket that has just taken off from from the Earth's surface.
The rocket's displacement function at this point has been empirically established s(t) = 2t + 1.2 t2
(i) Find v(t) the rocket's velocity as a function ot time.
(ii) Find the rocket's acceleration and state why it is constant?
(iii) What is the rocket's velocity 0.5s after its displacement function has been established.
(iv) Explain why, in this context, that it probably would not be correct to evaluate this displacement function for long time intervals. (v) Write down a possible displacement function at 5 km, say, above the surface of the Earth.

The non-relativity of light speed

To illustrate the time dilation expression ...
Time dilation expression
Suppose the denominator (the bottom line) of the expression equals 1/3. This means if Olivia measured the period of a pendulum on the passing train and got 3.0s, then Thomas would record 1.0s, thus his clock is clearly running slower. This is an unexpected result, but it has been verified by experiment and is a direct consequence of the constancy of the speed of light for all observers. You are invited to calculate the speed a vehicle must be travelling in order for the time intervals to increase by a factor of three (ie for events to take 3 times longer than those times measured by a stationary observer).

The Twin Paradox

Suppose Thomas and Olivia are twins at a Launch pad on Earth. Thomas jumps in his aircraft and flies off at a speed of 0.5 c. This means (remember to verify this) time periods on his aircraft as measured by Olivia will be &radic3/2 (= 0.86) x her readings. Now if the spacecraft goes on a journey for ten years as measured by Olivia, Thomas' will have recorded on his watch ten years too, but his ten years are not the same as his sister's ten years. In fact because his clock ran slow, he only aged by 8.6 years. This is not the Twin Paradox, it is merely an interesting and factually correct curiosity. Thomas could say to his sister, yes but you also moved at 0.5 c, relative to me and if the mirror and light pulse was next to you as I passed by the expression would be ...
Time dilation expression, with Olivia seen to be in relative motion wrt to Thomas
This paradox is resolved by noting the situations for Thomas and Olivia are not identical in that Thomas had to accelerate and decelerate as he went out and came back. Olivia, on the other hand, was always at rest.

1) Kinematic relationships as derivatives.

The starting point for this derivation is to write instantaneous velocity and acceleration as derivatives and then use calculus to find the equations of motion. Equations of Motion derived using calculus

2) Working with the equations of motion

In problems we are normally presented with a displacement function and asked to find its associated velocity function. The following problem is about a rocket. The vehicle has taken off and its displacement at time t is given as s(t) = 1.2t + 2t2 (in the form s = ut + 1/2at2). Assume displacement is in meters and time is in seconds. The questions tend to ask three things.
1) Find an expression for the velocity of the rocket. 2) Find the value of the acceleration. 3) Find the velocity of the vehicle after a given time. In this case we shall use 2.0 s.
Equations of Motion example

Advanced Higher Physics .. Unit 1 Mechanics .. 1.1 Kinematic relationships and relativistic motion .. BR (2011)