Question

The position of a particle moving along the x axis may be determined from the expression x(t) = btu + ctv, where x will be in meters when t is in seconds. What will be the dimensions of b and c in this case if u = 8 and v = 7?

Answer 1

As per given equation we have

$x = bt^u + ct^v$

now as per the dimensional analysis we can say that dimension of right side of equation must be equal to left side of the equation

now as per left side of equation its dimension is same as length or meter

now we can say it should be meter on right side also

$bt^u = M^0L^1T^0$

$b*T^8 = M^0L^1T^0$

$b = M^0L^1T^{-8}$

similarly for other term we have

$ct^v = M^0L^1T^0$

$c*T^7 = M^0L^1T^0$

$c = M^0L^1T^{-7}$

so above are the dimensions of b and c

Answer 2

The dimension of b and c is $b = M^0L^1T^{-8$ and $c = M^0L^1T^{-7$

Calculation of dimension:

Since the x axis may be determined from the expression that shown below.

$x = bt^u + ct^v$

Here,  x will be in meters when t is in seconds

Now

Here the dimension of the right side should be equivalent to the left side.

According to the left side the dimension should be same like length or meter so same goes for the right side

So,

$bt^u = M^0L^1T^0\\\\b\times T^8 = M^0L^1T^0\\\\b = M^0L^1T^{-8}$

Now for another one it is

$ct^v = M^0L^1T^0\\\\c\times T^7 = M^0L^1T^0\\\\c = M^0L^1T^{-7$

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