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Computing Apparent Mass From Rest Mass and Speed

Date: 05/27/2002 at 14:11:09
From: Dustin
Subject: The actual equation to find the mass of a moving object

I've been trying the following equation that I found that is supposed 
to find the mass of a moving object....

   Apparent Mass = Rest Mass * 1/sqrt(1 - v2/c2)

   v = speed
   c = light speed

Could you tell me if this is the correct equation?  If it is, could 
you show me an example where

  Rest Mass = 10
  v = 20

I am going to assume v is in mph, so 20 mph, and rest mass is 10 
grams. I would really appreciate an example that shows the value you 
plug in for lightspeed and the steps to solve this equation.  Thank 
you very much!

Date: 05/27/2002 at 19:58:29
From: Doctor Rick
Subject: Re: The actual equation to find the mass of a moving object

Hi, Dustin.

That's the correct formula (although we usually write "v^2" for v 
squared so it doesn't look like v times 2). You're using a strange 
mixed set of units: metric mass and English system for distance. That 
won't be a problem, though, as long as you express c, the speed of 
light, in the same units as v, because these units will cancel out, 
leaving a dimensionless number inside the square root. 

The speed of light is approximately 3*10^8 km/s (that's 300,000,000 
km per second). Convert this to miles/hour and use it for c.

- Doctor Rick, The Math Forum 

Date: 05/27/2002 at 20:57:31
From: Dustin
Subject: (Continued)

Ok, thank you for finally clearing what the value for light speed is 
for me. But I see one problem with this equation.  The moving mass 
comes out to be exactly the same no matter what numbers you plug in 
for the rest mass and for the speed.  How fast does an object have to 
be going for it to change mass?

Either I am doing something extremely wrong, or this equation gives 
you basically the same number answer (19.9999 = Apparent mass when 
Rest Mass = 20 and speed = 20 when I plug in numbers, along with the 
value you've given me for lightspeed).  Its always coming out the 
same.  Why?    Thank you.

Date: 05/28/2002 at 08:28:31
From: Doctor Rick
Subject: Re: (Continued)

Hi, Dustin.

It doesn't come out *exactly* the same, but the difference is so many 
decimal places to the right that your calculator or whatever doesn't 
show it. At speeds like 20 mph, Newton's laws describe the world very 
well, and they don't include any mass increase with velocity. As a 
rule of thumb, "relativistic" effects such as mass increase don't 
start to become noticeable until the velocity is about 1/10 the speed 
of light -- that is, 3*10^7 (30,000,000) km/sec, or 6.7*10^10 
(67,000,000,000) mph. That's a lot more than 20 mph!

- Doctor Rick, The Math Forum 
Associated Topics:
High School Physics/Chemistry

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