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Question 19M.3.HL.TZ2.22

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Date May 2019 Marks available [Maximum mark: 5] Reference code 19M.3.HL.TZ2.22
Level HL Paper 3 Time zone TZ2
Command term Estimate, Justify, Show that Question number 22 Adapted from N/A
22.
[Maximum mark: 5]
19M.3.HL.TZ2.22

The homogeneous model of the universe predicts that it may be considered as a spherical cloud of matter of radius r and uniform density ρ. Consider a particle of mass m at the edge of the universe moving with velocity v and obeying Hubble’s law.

(a)

Justify that the total energy of this particle is E = 1 2 m v 2 − 4 3 π G r r 2 m .

[2]

Markscheme

total energy=kinetic energy+potential energy

OR

total energy= 1 2 m v 2 − G M m r ✔

substitution of M = 4 3 π r 3 ρ ✔

«Hence answer given»

Answer given so for MP2 look for clear evidence that MUniverse ( 4 3 π r 3 ρ ) is stated and substituted.

Examiners report

The vast majority of the candidates could state that the total energy is equal to the sum of the kinetic and potential energies but quite a few did not use the correct formula for the gravitational potential energy. The formula for the mass of the sun was usually correctly substituted.

(b)

At critical density there is zero total energy. Show that the critical density of the universe is:  r c = 3 H 0 2 8 π G .

[2]

Markscheme

substitutes H0r for v ✔

«total energy = 0»

1 2 m H 0 2 r 2 = 4 3 π G ρ r 2 m   ✔

«hence ρc = 3 H 0 2 8 π G »

Answer given, check working carefully.

Examiners report

This was a relatively easy demonstration given the equation in 22a. However many candidates did not show the process followed in a coherent manner that could be understood by examiners.

(c)

The accepted value for the Hubble constant is 2.3 × 10−18 s−1. Estimate the critical density of the universe.

[1]

Markscheme

9.5 × 10−27 « kgm–3» ✔

Examiners report

The question was well answered by many candidates.

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