Question

The velocity v of a particle depends upon time t , according to the equation v =$a+bt+ \frac{c}{d+t}$

Write the dimensions of a , b , c, and d.

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Write the dimensions of a/b in the relation F=a√x + bt² where F is the force , x is the distance and t is time.

Answer 1

Two quantities can be added or subtracted if they have the same units. The result obtained by addition or subtraction has the same unit as that of the quantities being added or subtracted.

First Part:
By the above logic:
Unit of a = same as that of velocity = m/s

Unit of bt = same as that of velocity 
Therefore, unit of b = $\frac{unit of velocity}{unit of time}$ =$m/s^2$

Unit of d = same as that of time = sec

Unit of $\frac{c}{d+t}$ = same as that of velocity
Therefore, unit of c = unit of velocity * unit of (d+t)
unit of (d+t) = unit of t = unit of d = sec
Hence unit of c = unit of velocity *  unit of time = m

Second Part:
Unit of a$\sqrt{x}$ = Unit of Force
Therefore, unit of a = $\frac{Unit\ of\ force}{Unit\ of\ \sqrt{x} } = \frac{Kg\ m\ /s^2}{ \sqrt{m} }=\frac{Kg \ \sqrt{m}}{s^2}$

Unit of $bt^2$ = Unit of force
Therefore, unit of b = $\frac{Unit\ of\ force}{(Unit\ of\ time)^2 } = \frac{Kg\ m\ /s^2}{ s^{2} } = \frac{Kg \ m}{s^4}$

Unit of a/b = Unit of a / Unit of b
=  $\frac{Kg\ \sqrt{m} \ /s^2}{Kg\ m \ /s^4} = \frac{Kg \ s^2}{ \sqrt{m} }$

Answer 2

Answer:

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