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Video: Determining the Velocity of an Object of Known Mass from Its Relativistic Momentum

Ed Burdette

What is the velocity of an electron that has a relativistic momentum of 3.040 × 10⁻²¹ kg⋅m/s? Electron rest mass is 9.109 × 10⁻³¹ kg.

04:28

Video Transcript

What is the velocity of an electron that has a relativistic momentum of 3.040 times 10 to the negative twenty-first kilograms meters per second. Electron rest mass is 9.109 times 10 to the negative thirty-first kilograms.

In this problem, we’ll assume that 𝑐, the speed of light, is exactly 3.00 times 10 to the eighth meters per second. In this statement, we’re told the relativistic momentum of an electron, that it’s 3.040 times 10 to the negative twenty-first kilograms meters per second. We’ll call that value 𝑝. And we’re also told that the electron rest mass is 9.109 times 10 to the negative thirty-first kilograms. We’ll call that 𝑚 sub zero, where zero represents rest. We want to know the velocity of the electron that has this momentum. We’ll call that velocity 𝑣.

To begin our solution, let’s start by recalling the relationship for relativistic momentum. The relativistic momentum 𝑝 of an object is equal to 𝛾 times the rest mass of that object times its velocity 𝑣. Notice how similar this is to nonrelativistic momentum where 𝑝, the momentum, is equal to simply 𝑚 times 𝑣, the mass of an object times its velocity. So the difference here is 𝛾, where 𝛾 is defined as one divided by the square root of one minus the particle speed 𝑣 squared divided by the speed of light squared.

When we write out the equation for relativistic momentum of this electron, 𝑝 is equal to the rest mass of the electron times its velocity divided by the square root of one minus its velocity squared over 𝑐 squared. We want to rearrange this equation algebraically for the velocity 𝑣.

Let’s start by squaring both sides of the equation. When we do that, the square root in the denominator cancels out with the squared term. Then if we multiply both sides by one minus 𝑣 squared over 𝑐 squared, that term cancels on the right side of our equation. If we then multiply through the 𝑝 squared term on the left side of our equation, we can then add 𝑝 squared times 𝑣 squared over 𝑐 squared to both sides which cancels that term on the left-hand side of our equation. We then factor out 𝑣 squared from the right-hand side of our equation. We then divide both sides of the equation by the quantity 𝑚 zero squared plus 𝑝 squared over 𝑐 squared which cancels out that term from the right-hand side of the equation. And finally, we take the square root of both sides. This cancels out the square root and squared term on the right side leaving us with an expression for the velocity 𝑣.

𝑣 is equal to the electrons relativistic momentum 𝑝 divided by the square root of its rest mass squared plus its momentum squared over 𝑐 squared where we treat 𝑐 as exactly 3.00 times 10 to the eighth meters per second. When we enter the values for 𝑝, 𝑚 sub zero, and 𝑐 into this equation, then we are now able to solve for 𝑣 by entering these values into our calculator. Notice that the units that we find when we calculate these values end up being meters per second, just as we would expect for a velocity.

And we find a velocity of 2.988 times 10 to the eighth meters per second. This is the speed, very close to the speed of light 𝑐, of an electron that has this relativistic momentum.