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1. Lorentz contraction and time...

1. Lorentz contraction and time dilation are two phenomena arising from

Einstein's special theory of relativity. Briefly explain these two phenomena

and why they are not observed in daily life (in a few sentences). Illustrate

your answer by considering the magnitude of the factor

sqrt1/v^2/c^2

for an airliner flying at a speed of 0.20 km s^-1

You may wish to make use of the approximation

sqrt 1-v^2/c^2~1-v^2/2c^2

which is very good for v^2/c^2 << 1, and consider the value of the term

v^2/2c^2

(b) Assume that with modern technology a journey to the planet Mars and back

to Earth could be undertaken in one Earth-year at a speed of 1.00 × 10

^4 m s^-1.

By how much less would the astronauts have aged, compared with their

Earth-bound colleagues, after undertaking the journey? You can use the

approximation given in (a)

(c) For this part of the question you will calculate relativistic energies and

momenta for bodies travelling at a substantial fraction of the speed of light

and so the approximation given in (a) cannot be used.

Keep three significant figures in your working but round down appropriately

when giving your final answer in part (vi).

(i) Determine the rest mass energy and the (total) relativistic energy (in

joules) of a body of mass 5.0 kg travelling at a speed of 1.5 × 10^8

m s^-1

Determine also the relativistic momentum (in kg m s^-1) of the body.

Now suppose the body collides with, and sticks to, a stationary body also of

rest mass 5.0 kg.

(ii) Determine the sum total of the relativistic energy of the two bodies

before they collide, and state their total relativistic momentum.

(iii) Let M be the rest mass and vC the speed of the composite body after the

collision. (Both M and vC are unknowns at this stage.) Write down

expressions for the (total) relativistic energy and the relativistic

momentum of the composite particle in terms of M and vC.

(iv) Write down an equation expressing the conservation of relativistic energy

for the collision.

(v) Write down an equation expressing the conservation of relativistic

momentum.

(vi) Hence determine the speed vC. (Hint: you should be able to eliminate M

and the unknown Lorentz factor from the two equations you obtained in

(iv) and (v) above, to obtain a single equation in vC which you can solve.)

Einstein's special theory of relativity. Briefly explain these two phenomena

and why they are not observed in daily life (in a few sentences). Illustrate

your answer by considering the magnitude of the factor

sqrt1/v^2/c^2

for an airliner flying at a speed of 0.20 km s^-1

You may wish to make use of the approximation

sqrt 1-v^2/c^2~1-v^2/2c^2

which is very good for v^2/c^2 << 1, and consider the value of the term

v^2/2c^2

(b) Assume that with modern technology a journey to the planet Mars and back

to Earth could be undertaken in one Earth-year at a speed of 1.00 × 10

^4 m s^-1.

By how much less would the astronauts have aged, compared with their

Earth-bound colleagues, after undertaking the journey? You can use the

approximation given in (a)

(c) For this part of the question you will calculate relativistic energies and

momenta for bodies travelling at a substantial fraction of the speed of light

and so the approximation given in (a) cannot be used.

Keep three significant figures in your working but round down appropriately

when giving your final answer in part (vi).

(i) Determine the rest mass energy and the (total) relativistic energy (in

joules) of a body of mass 5.0 kg travelling at a speed of 1.5 × 10^8

m s^-1

Determine also the relativistic momentum (in kg m s^-1) of the body.

Now suppose the body collides with, and sticks to, a stationary body also of

rest mass 5.0 kg.

(ii) Determine the sum total of the relativistic energy of the two bodies

before they collide, and state their total relativistic momentum.

(iii) Let M be the rest mass and vC the speed of the composite body after the

collision. (Both M and vC are unknowns at this stage.) Write down

expressions for the (total) relativistic energy and the relativistic

momentum of the composite particle in terms of M and vC.

(iv) Write down an equation expressing the conservation of relativistic energy

for the collision.

(v) Write down an equation expressing the conservation of relativistic

momentum.

(vi) Hence determine the speed vC. (Hint: you should be able to eliminate M

and the unknown Lorentz factor from the two equations you obtained in

(iv) and (v) above, to obtain a single equation in vC which you can solve.)

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