Speed and energy of particles – Newtonian calculation
Question 10S: Short Answer		

In these questions you will calculate the speed of charged particles accelerated by different potential differences. You will find where relativistic effects start to be important.
Relationships to use

The kinetic energy given to a particle with charge q accelerated through a potential difference V is:

Ekinetic = qV

The speed of the particle, given its mass m can be found from the Newtonian relationship for kinetic energy:

This calculation of speed is only valid for speeds much smaller than the speed of light.
Data to use
Charge on electron = 1.6 ´ 10–19 C	Mass of electron = 9.1 ´ 10–31 kg	Speed of light = 3.0 ´ 108 m s–1

Speed from kinetic energy
1.	Given the equation for kinetic energy Ekinetic = ½ mv 2, show that the speed v can be found from

2.	Calculate the kinetic energy of a 500 kg car travelling at 20 m s–1.

3.	Calculate the speed of the same car, if its kinetic energy is double the previous value.

4.	Calculate the kinetic energy of an electron accelerated through a p.d. of 1000 V, typical of a school laboratory electron tube. Use the value of the energy to calculate the speed of the electron. What percentage of the speed of light is the speed you have calculated?

5.	Repeat the calculations of kinetic energy and speed for electrons accelerated through a p.d. of 64 000 V in a dental x-ray tube. What fraction of the speed of light does this speed represent?
Speed and energy of particles – Newtonian calculation
Question 10S: Short Answer		

In these questions you will calculate the speed of charged particles accelerated by different potential differences. You will find where relativistic effects start to be important.
Relationships to use

The kinetic energy given to a particle with charge q accelerated through a potential difference V is:

Ekinetic = qV

The speed of the particle, given its mass m can be found from the Newtonian relationship for kinetic energy:

This calculation of speed is only valid for speeds much smaller than the speed of light.
Data to use
Charge on electron = 1.6 ´ 10–19 C	Mass of electron = 9.1 ´ 10–31 kg	Speed of light = 3.0 ´ 108 m s–1

Speed from kinetic energy
1.	Given the equation for kinetic energy Ekinetic = ½ mv 2, show that the speed v can be found from

2.	Calculate the kinetic energy of a 500 kg car travelling at 20 m s–1.

3.	Calculate the speed of the same car, if its kinetic energy is double the previous value.

4.	Calculate the kinetic energy of an electron accelerated through a p.d. of 1000 V, typical of a school laboratory electron tube. Use the value of the energy to calculate the speed of the electron. What percentage of the speed of light is the speed you have calculated?

5.	Repeat the calculations of kinetic energy and speed for electrons accelerated through a p.d. of 64 000 V in a dental x-ray tube. What fraction of the speed of light does this speed represent?

6.	In fact the calculation in question 5 is not quite correct. The speed cannot increase indefinitely because no material particle can reach the speed of light. But, if the calculation were correct, at what potential difference does the calculation say the electron would reach the speed of light?

This graph compares the Newtonian and relativistic calculations of kinetic energy. You can see that the two calculations are extremely close up to speeds of about 20% of the speed of light.

7.	Find the potential difference through which an electron has to be accelerated to reach 20% of the speed of light.

6.	In fact the calculation in question 5 is not quite correct. The speed cannot increase indefinitely because no material particle can reach the speed of light. But, if the calculation were correct, at what potential difference does the calculation say the electron would reach the speed of light?

This graph compares the Newtonian and relativistic calculations of kinetic energy. You can see that the two calculations are extremely close up to speeds of about 20% of the speed of light.

7.	Find the potential difference through which an electron has to be accelerated to reach 20% of the speed of light.