Why do square of a number and square roots of a number so integral to important equations?
Example: Most famous equation the E=mc^2
Why not E=mc^1.9999999999 or E=mc^2.0000000001 ?
Why is EXACTLY the square root so important?
Example: Most famous equation the E=mc^2
Why not E=mc^1.9999999999 or E=mc^2.0000000001 ?
Why is EXACTLY the square root so important?
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Good question. In general, most equations from physics will have integer exponents....or at worst rational exponents. This follows from something called dimensional analysis.
What is dimensional analysis? It's just fancy-speak for "making sure your units match up". If I asked you to add 4 kimometers plus 3 seconds, you couldn't, right? Well, dimensional analysis makes sure your units match up. The rules are simple:
1) You can multiply any two units to make a new unit (ie meter times meter gives m^2, or a square meter)
2) You can divide any two units to make a new unit (ie a meter divided by a second gives a m/s, a unit of speed or velocity).
3) You can ONLY add and subtract like units, or units that can be converted to like units (ie you can add 1000m + 1cm, but you can't subtract 1 km - 3 hours.)
4) If an equation says two things are equal, their units must be equal too. Thus d = rt works, since distance can be measured in meters, and rt can be measured in (meters/seconds) * seconds = meters.
So, specifically for E = mc^2, you have energy on the left. You'll learn in physics that:
1) Energy is equal to work, which is equal to the product (or dot product) of force times distance. So E = Fd.
2) Force is equal to mass times acceleration, newton's law. So F = ma. Thus, energy must have the same units as F * d, or m * a * d
3) Acceleration is the rate of change of velocity with respect to time...or a = v/t. And velocity is the rate of change of distance with respect to time, or v = d/t. Thus, acceleration will have units of meters per second squared, or m/s^2.
Putting all those together, you can see that energy will have the same units as m * a * d. Since mass is kg, acceleration is m/s^2, and distance is m, energy has units of kg * m/s^2 * m, or kg * m^2 / s^2.
What is dimensional analysis? It's just fancy-speak for "making sure your units match up". If I asked you to add 4 kimometers plus 3 seconds, you couldn't, right? Well, dimensional analysis makes sure your units match up. The rules are simple:
1) You can multiply any two units to make a new unit (ie meter times meter gives m^2, or a square meter)
2) You can divide any two units to make a new unit (ie a meter divided by a second gives a m/s, a unit of speed or velocity).
3) You can ONLY add and subtract like units, or units that can be converted to like units (ie you can add 1000m + 1cm, but you can't subtract 1 km - 3 hours.)
4) If an equation says two things are equal, their units must be equal too. Thus d = rt works, since distance can be measured in meters, and rt can be measured in (meters/seconds) * seconds = meters.
So, specifically for E = mc^2, you have energy on the left. You'll learn in physics that:
1) Energy is equal to work, which is equal to the product (or dot product) of force times distance. So E = Fd.
2) Force is equal to mass times acceleration, newton's law. So F = ma. Thus, energy must have the same units as F * d, or m * a * d
3) Acceleration is the rate of change of velocity with respect to time...or a = v/t. And velocity is the rate of change of distance with respect to time, or v = d/t. Thus, acceleration will have units of meters per second squared, or m/s^2.
Putting all those together, you can see that energy will have the same units as m * a * d. Since mass is kg, acceleration is m/s^2, and distance is m, energy has units of kg * m/s^2 * m, or kg * m^2 / s^2.
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