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Suppose we are told that the position of an object (x) moving with an acceleration a is proportional to some power of a, say an and some power of t, say tm. Determine the values of n and m and write the simplest form of an equation for the position.

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onyebuchinnaji

(a) The value of n is 1 and the value of m is 2

(b) The simplest form of an equation for the position is [tex]at^2[/tex]

The given expression:

[tex]x = a^nt^m[/tex]

To find:

  • the values of a values of n and m
  • simplest form of the equation

The values of n and m will be calculated using the dimensions of accelerations, position and time.

  • The dimension of acceleration, [tex]a = LT^{-2}[/tex]
  • The dimension of position, x = L
  • The dimension of time, t = T

(a) the values of n and m are calculated as follows:

[tex]x = a^n t^m\\\\ L = [LT^{-2}]^n [T]^m\\\\L = [L]^n [T^{-2n}][T]^m\\\\L = [L]^n[T^{m-2n}]\\\\L^1T^0 = [L]^n[T^{m-2n}]\\\\L^1 = L^n\\\\n = 1\\\\T^0 = T^{m-2n}\\\\0 = m - 2n\\\\0 = m- 2(1)\\\\0 = m- 2\\\\m = 2[/tex]

(b) The simplest form of an equation for the position can be written as;

[tex]x = a^n t^m\\\\x = a^1 t^2\\\\x = at^2[/tex]

Learn more here: https://brainly.com/question/14639033

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