Video: Calculating the Relativistic Momentum

Find the momentum of a helium nucleus having a mass of 6.68 Γ— 10⁻²⁷ kg that is moving at 0.200𝑐.

01:53

Video Transcript

Find the momentum of a helium nucleus having a mass of 6.68 times 10 to the negative 27th kilograms that is moving at 0.200𝑐.

We can call the given mass, 6.68 times 10 to the negative 27th kilograms, π‘š. And we can call the nucleus’ speed, 0.200𝑐, 𝑣. We want to solve for the nucleus’s momentum. We’ll call that 𝑝. Now, starting off, we may not know at first whether to use the classical or the relativistic formulation for momentum. The clue in the problem statement that tells us to use relativistic is the fact that the speed of this nucleus is comparable to the speed of light. That means relativistic effects will be important to include.

Relativistic momentum 𝑝 is equal to 𝛾 times rest mass, π‘š sub zero, times 𝑣, where 𝛾 is one over the square root of one minus 𝑣 squared over 𝑐 squared. So, 𝑝 equals π‘šπ‘£ over the square root of one minus 𝑣 squared over 𝑐 squared, where 𝑣 is given to us in the problem statement as two-tenths the speed of light 𝑐. And π‘š also is given to us in the statement. So, we’re now ready to plug in to solve for momentum.

When we enter our values to this equation, the factor of the speed of light 𝑐 under the square root sign cancels out. But in the numerator, it remains. In this exercise, we’ll treat 𝑐 as exactly 3.00 times 10 to the eighth meters per second. When we make that substitution and then enter these values into our calculator, we find that 𝑝 is 4.09 times 10 to the negative 19th kilograms meters per second. That’s the relativistic momentum of this helium nucleus.

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