Worksheet: Relativistic Energy

In this worksheet, we will practice calculating the kinetic energy of objects moving at speeds large enough such that relativistic effects need to be taken into account.

Q1:

A Van de Graaff accelerator utilizes a 50.0 MV potential difference to accelerate charged particles such as protons. The kinetic energy provided by such a large potential difference is sufficiently great that relativistic effects need to be taken into account when finding the velocity of accelerated particles.

What is the velocity of a proton accelerated by such a potential?

  • A 0 . 3 1 4 𝑐
  • B 0 . 3 0 6 𝑐
  • C 0 . 3 0 2 𝑐
  • D 0 . 3 1 0 𝑐
  • E 0 . 3 1 8 𝑐

What is the velocity of an electron accelerated by such a potential?

  • A 0 . 9 9 9 9 9 𝑐
  • B 0 . 9 9 9 9 0 𝑐
  • C 0 . 9 9 9 9 5 𝑐
  • D 0 . 9 9 9 9 2 𝑐
  • E 0 . 9 9 9 9 7 𝑐

Q2:

What is the rest energy of an electron, given its mass is 9.11×10 kg?

Q3:

Determine the difference in the rest masses of a proton and a neutron from their rest energies. Use a value of 938.3 MeV for proton rest energy and 939.6 MeV for the neutron rest energy.

  • A 3 . 3 × 1 0    kg
  • B 2 . 3 × 1 0    kg
  • C 2 . 0 × 1 0    kg
  • D 3 . 0 × 1 0    kg
  • E 1 . 3 × 1 0    kg

Q4:

What is the velocity of an electron that has a kinetic energy of 0.750 MeV?

  • A 0 . 9 1 9 𝑐
  • B 0 . 9 0 5 𝑐
  • C 0 . 9 1 4 𝑐
  • D 0 . 9 2 3 𝑐
  • E 0 . 9 1 0 𝑐

Q5:

Suppose that the speed of light in a vacuum is only 45.0 m/s.

Calculate the relativistic kinetic energy of a car of mass 1.00×10 kg that has a velocity of 30.0 m/s.

  • A 6 . 6 2 × 1 0  J
  • B 7 . 3 1 × 1 0  J
  • C 6 . 3 9 × 1 0  J
  • D 6 . 0 6 × 1 0  J
  • E 6 . 9 2 × 1 0  J

Find the ratio of the car’s relativistic kinetic energy to its classical kinetic energy.

  • A 1 . 6 2 ∶ 1
  • B 1 . 5 4 ∶ 1
  • C 1 . 7 7 ∶ 1
  • D 1 . 3 1 ∶ 1
  • E 1 . 4 3 ∶ 1

Q6:

A 𝜋-meson has a rest mass of 135 MeV. The proper lifetime of the 𝜋-meson is 8.40×10 s and an observer in a laboratory measures its lifetime as 1.40×10 s. What is the kinetic energy of the 𝜋-meson as measured by the observer?

Q7:

K mesons have an average lifetime in their rest frame of 1.24×10 s. Plans for an accelerator that produces a secondary beam of K mesons to scatter from nuclei, for the purpose of studying the strong force, call for them to have a kinetic energy of 500 MeV.

What would the relativistic quantity 𝛾=11− be for these particles?

What would be the average lifetime of these particles, as measured by a laboratory based observer?

  • A 2 . 6 4 × 1 0   s
  • B 2 . 5 0 × 1 0   s
  • C 2 . 2 0 × 1 0   s
  • D 2 . 7 2 × 1 0   s
  • E 2 . 3 5 × 1 0   s

How far would these particles travel during their average lifetime, as measured by a laboratory based observer?

Q8:

Beta decay is a type of nuclear decay in which an electron is emitted from an atomic nucleus. If the electron is given 6.000 MeV of kinetic energy, what is its velocity?

  • A 0 . 9 5 6 3 𝑐
  • B 0 . 9 7 8 5 𝑐
  • C 0 . 9 9 6 9 𝑐
  • D 0 . 9 6 5 2 𝑐
  • E 0 . 9 9 9 9 𝑐

Q9:

Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We observe 1,895 nm wavelength electromagnetic radiation but know that the emitted wavelength was 1,870 nm. What is the relative speed of the gas?

  • A 0 . 0 2 3 5 𝑐
  • B 0 . 0 1 6 8 𝑐
  • C 0 . 1 2 5 6 𝑐
  • D 0 . 1 5 8 0 𝑐
  • E 0 . 0 1 3 3 𝑐

Q10:

A neutral kaon is a particle that decays into two 𝜋-mesons. The kaon has a rest mass energy of 497.6 MeV and muons have a rest mass energy of 105.7 MeV. Suppose the kaon is at rest and all the missing mass goes into the muons’ kinetic energy. Assuming energy is shared equally between the two muons, how fast will the muon move?

  • A 0 . 8 5 4 6 𝑐
  • B 0 . 9 0 5 3 𝑐
  • C 0 . 6 7 4 1 𝑐
  • D 0 . 9 6 2 9 𝑐
  • E 0 . 9 2 9 3 𝑐

Q11:

A positron is an antimatter version of the electron, having the same mass. When a positron and an electron meet, they annihilate converting all their mass into energy.

Find the energy released, assuming negligible kinetic energy before the annihilation.

If the annihilation energy released is given to a proton in the form of kinetic energy, what is the proton’s velocity?

  • A 0 . 0 4 7 𝑐
  • B 0 . 1 1 5 𝑐
  • C 0 . 0 5 2 𝑐
  • D 0 . 1 2 1 𝑐
  • E 0 . 0 3 3 𝑐

If the annihilation energy released is given to an electron in the form of kinetic energy, what is the electron’s velocity?

  • A 0 . 8 8 9 𝑐
  • B 0 . 8 1 7 𝑐
  • C 0 . 9 1 2 𝑐
  • D 0 . 9 4 3 𝑐
  • E 0 . 8 5 4 𝑐

Q12:

The big bang that began the universe is estimated to have released 4×10 J of energy. How many stars could be created by half of that energy if the average mass of a star is 5.00×10 kg?

  • A 1 × 1 0   stars
  • B 4 × 1 0   stars
  • C 2 × 1 0   stars
  • D 4 × 1 0   stars
  • E 8 × 1 0   stars

Q13:

A matter with a mass of 2.00 kg is converted to energy.

Calculate the energy released by the destruction of the matter.

  • A 6 0 0 × 1 0  J
  • B 1 . 8 0 × 1 0   J
  • C 1 . 8 0 × 1 0   J
  • D 2 5 0 × 1 0   J
  • E 5 . 4 0 × 1 0  J

How many kilograms could be lifted to a 20.0 km height by the energy released by the destruction of the matter?

  • A 9 . 8 1 × 1 0   kg
  • B 9 . 1 8 × 1 0   kg
  • C 9 . 8 1 × 1 0   kg
  • D 9 . 1 8 × 1 0  kg
  • E 9 . 1 8 × 1 0   kg

Q14:

Find the kinetic energy, in mega-electron volts, of a charged pion with a measured lifespan of 5.96×10 s, given that its rest energy is 139.6 MeV and its proper lifespan is 2.60×10 s.

Q15:

When an electron and a positron collide in a linear accelerator, they each have a kinetic energy of 48.5 GeV. What is the total collision energy available? Assume that the energy produced from the annihilation of the two particles is negligible.

  • A 146 GeV
  • B 48.5 GeV
  • C 24.3 GeV
  • D 20.6 GeV
  • E 97.0 GeV

Q16:

An electron’s rest energy can be measured in either joules or electron volts.

Find the rest energy of an electron in joules, precise to three significant figures.

  • A 2 . 7 3 × 1 0    J
  • B 3 . 5 2 × 1 0    J
  • C 8 . 2 0 × 1 0    J
  • D 7 . 6 0 × 1 0    J
  • E 8 . 5 0 × 1 0    J

Find the rest energy of an electron in mega-electron volts, precise to three significant figures.

  • A 1 . 7 1 × 1 0   MeV
  • B 2 2 . 0 × 1 0   MeV
  • C 5 1 2 × 1 0   MeV
  • D 5 3 1 × 1 0  MeV
  • E 0 . 4 7 5 × 1 0  MeV

Q17:

A particle has a mass equal to 20 u. If this mass is converted completely into energy, how much energy is released? Express your answer in megaelectron-volts.

  • A 2 × 1 0  MeV
  • B 2 × 1 0  MeV
  • C 6 × 1 0   MeV
  • D 2 × 1 0   MeV
  • E 6 × 1 0  MeV

Q18:

A cosmic ray neutron has a velocity of 0.240 c relative to Earth.

What is the neutron’s total energy in mega-electron volts?

What is the neutron’s momentum?

  • A 1 . 3 8 × 1 0    kg⋅m/s
  • B 1 . 5 8 × 1 0    kg⋅m/s
  • C 2 . 3 2 × 1 0    kg⋅m/s
  • D 1 . 2 4 × 1 0    kg⋅m/s
  • E 2 . 5 3 × 1 0    kg⋅m/s

Q19:

The energy released from the fission of uranium by a Hiroshima-sized bomb containing 1.00 kg of uranium is 4.2×10 J.

Calculate the mass converted to energy by the fission reaction.

  • A 4 7 × 1 0   kg
  • B 4 . 7 × 1 0   kg
  • C 4 7 × 1 0   kg
  • D 4 7 × 1 0   kg
  • E 4 7 × 1 0   kg

What is the ratio of mass of the uranium destroyed by the fission reaction to the original mass of the uranium in the bomb?

  • A 4 . 7 × 1 0  
  • B 4 . 7 × 1 0  
  • C 4 . 7 × 1 0  
  • D 4 . 7 × 1 0  
  • E 4 . 7 × 1 0  

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