Worksheet: Relativistic Kinetic Energy

In this worksheet, we will practice calculating the relativistic kinetic energy of massive objects moving relative to an observer.

Q1:

The graph shows the kinetic energy, EK, against the speed, 𝑣, of two objects, 𝐴 and 𝐡. The relativistic kinetic energy of object 𝐴 is shown by the red line, whereas the classical kinetic energy of object 𝐡 is shown by the blue line. Object 𝐴 has a rest mass π‘šοŒ  and object 𝐡 has a rest mass π‘šοŒ‘. The speed of light, 𝑐, is defined to be 1.

Which object has a greater rest mass?

  • AObject 𝐡
  • BObject 𝐴

At a speed of 0.8𝑐, the classical kinetic energy of object 𝐡 has the same value as the relativistic kinetic energy of object 𝐴. The formula π‘š=2π‘šο€Ώο„ž1𝑣+1π‘£βˆ’π‘£βˆ’1π‘£ο‹οŒ‘οŒ οŠͺοŠͺ can be used to work out the rest mass of object 𝐡 from the rest mass of object 𝐴, where 𝑣=0.8𝑐. If π‘š=1 kg, what is the rest mass of object 𝐡?

  • A6216 kg
  • B1225 kg
  • C7516 kg
  • D2524 kg
  • E2512 kg

Q2:

A spaceship moves at a speed of 0.1𝑐 relative to Earth. The rest mass of the spaceship is 200,000 kg. What is the relativistic kinetic energy of the spaceship in the rest frame of Earth? The spaceship is far enough away from Earth such that the effect of Earth’s gravity can be ignored. Give your answer to 3 significant figures.

  • A9.07Γ—10 J
  • B2.00Γ—10 J
  • C1.81Γ—10 J
  • D1.82Γ—10 J
  • E9.73Γ—10 J

Q3:

Two spaceships are moving through deep space. The relativistic kinetic energy of spaceship A relative to spaceship B is 4.51Γ—10 J. The speed of spaceship A relative to spaceship B is 0.05𝑐. What is the mass of spaceship A? Give your answer to 3 significant figures.

Q4:

An electron is accelerated in a linear particle accelerator to a speed of 0.99𝑐. The rest mass of an electron is 511/𝑐keV. What is the relativistic kinetic energy of the electron? Ignore the Earth’s gravity. Give your answer in kilo-electron volts (keV) to 3 significant figures.

Q5:

A proton is accelerated in a linear particle accelerator to a speed of 0.5𝑐. The rest mass of a proton is 1.67Γ—10 kg. What is the relativistic kinetic energy of the proton? Ignore Earth’s gravity. Give your answer to 3 significant figures.

  • A2.33Γ—10 J
  • B5.01Γ—10J
  • C6.23Γ—10J
  • D1.74Γ—10 J
  • E2.13Γ—10 J

Q6:

A metal cube with a rest mass of 2 kg moves at a speed of 0.2𝑐 relative to an observer.

What would the classical kinetic energy of the cube be relative to the observer? Give your answer to 2 significant figures.

  • A3.7Γ—10 J
  • B7.2Γ—10 J
  • C9.0Γ—10 J
  • D1.8Γ—10 J
  • E3.6Γ—10 J

What is the relativistic kinetic energy of the cube relative to the observer? Give your answer to 3 significant figures.

  • A2.12Γ—10 J
  • B3.71Γ—10 J
  • C1.84Γ—10 J
  • D2.01Γ—10 J
  • E7.50Γ—10 J

What is the difference in the relativistic kinetic energy and the classical kinetic energy of the cube? Give your answer to 3 significant figures.

  • A1.76Γ—10 J
  • B1.12Γ—10οŠͺ J
  • C1.80Γ—10 J
  • D1.97Γ—10 J
  • E3.90Γ—10 J

Q7:

A positron moves along a linear particle accelerator with a speed of 0.998𝑐. Its relativistic kinetic energy is 7,570 keV. What is the mass of the positron? Give your answer in keV/π‘οŠ¨ to three significant figures.

  • A339/𝑐keV
  • B354/𝑐keV
  • C479/𝑐keV
  • D511/𝑐keV
  • E452/𝑐keV

Q8:

The graph shows the relativistic kinetic energy, 𝐸οŒͺ, of an object against the speed, 𝑣, of that object, where the speed of light, 𝑐, is defined to be 1. The object has a rest mass of 1 kg. Which of the following statements explains how this graph shows that it is impossible to accelerate a massive object to the speed of light?

  • AThe relativistic kinetic energy of an object cannot exceed π‘šπ‘οŠ¦οŠ¨, which on this graph has a value of 1.
  • BThe graph is asymptotic at 𝑣=1. This means that in order to accelerate a massive object to the speed of light, it would have to be given an infinite amount of kinetic energy.
  • CThe graph is asymptotic at 𝑣=1. This means that the object would have to have an imaginary mass to be able to travel at the speed of light.
  • DThe graph is asymptotic at 𝑣=1. This means that the object would have to have a negative mass to be able to travel at the speed of light.

Q9:

The graph shows both the classical kinetic energy and relativistic kinetic energy of an object against its speed. The object has a rest mass of 1 kg, and units of distance and time have been chosen such that the speed of light, 𝑐, has the value 1. The blue line shows the classical kinetic energy of the object and the red line shows the relativistic kinetic energy of the object.

Is the classical kinetic energy or the relativistic kinetic energy of the object greater at a speed of 0.6𝑐?

  • AThe classical kinetic energy is greater.
  • BThe relativistic kinetic energy is greater.

Which of the following statements is true?

  • AThe relativistic kinetic energy of the object is greater than the classical kinetic energy at any nonzero speed.
  • BThe classical kinetic energy of the object is greater than the relativistic kinetic energy at speeds greater than 0.2𝑐. At speeds less than 0.2𝑐, the classical kinetic energy and relativistic kinetic energy are equal.
  • CThe relativistic kinetic energy of the object is greater than the classical kinetic energy at speeds greater than 0.2𝑐. At speeds less than 0.2𝑐, the relativistic kinetic energy and classical kinetic energy are equal.
  • DThe classical kinetic energy of the object is greater than the relativistic kinetic energy at any nonzero speed.

What is the value of the classical kinetic energy of the object at 𝑣=1𝑐?

What is the value of the relativistic kinetic energy of the object at 𝑣=1𝑐?

  • A0
  • B10
  • C1
  • D0.5
  • EInfinite

Q10:

The graph shows the relativistic kinetic energy, 𝐸K, against the speed, 𝑣, of three objects, A, B, and C. The red line shows the relativistic kinetic energy of object A, the blue line object B, and the green line object C. Objects A, B, and C have rest masses π‘šA, π‘šB, and π‘šC respectively. The speed of light, 𝑐, is defined to be 1.

Which object has the greatest rest mass?

  • AObject C
  • BObject A
  • CObject B

For object B, at 𝑣=0.6, 𝐸=0.5K. What is the value of π‘šB?

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