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A famous relation in physics relates mov...

A famous relation in physics relates moving mass m with the rest mass `m_(0)` of a particle in terms of its speed v and the speed of light c. This relation first arose as a consequence of special relativity by Albert Einstein. A boy recalls the relation almost correctly but forgets where to put the constant c. He writes : `m=(m_(0))/((1-v^(2))^(1//2)).` Guess where to put the missing c.

Text Solution

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1 is a dimensionless number but `v^(2)` has a dimension `L^(2)T^(-2)` because [v] = `LT^(-1)` . So the expression (`1-v^(2))` does not have dimensional homogeneity. Here `v^(2)` should be replaced by a dimensionless quantity. As [c] = `LT^(-1)` and `[c^(2)]=L^(2)T^(-2),` we note that `(v^(2))/(c^(2))` is dimensionless. So the correct relation should be :
`m=(m_(0))/((1-v^(2)/c^(2))^(1//2))`
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