The velocity 'v' of a particle at time t is given by:

[tex]v = \frac{a}{t} + \frac{bt}{t^2 + c}.[/tex]

Determine the dimensions of a, b, and c respectively.

The dimensions are: a has a dimension of [L] (length), b has a dimension of [L] (length), and c has a dimension of [T^2] (time squared).
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Answered by Anonymous | 2025-07-04

To determine the dimensions of the constants a , b , and c in the given velocity function v = t a ​ + t 2 + c b t ​ , we need to consider the dimensions of each term in the equation. In physics, the dimensions of a physical quantity provide a way to describe the nature of the quantity in terms of basic dimensions such as length [ L ] , mass [ M ] , and time [ T ] .

Velocity ( v ):


The dimension of velocity is [ L ] [ T ] − 1 , i.e., length per time.


First Term ( t a ​ ) :


The dimension of t (time) is [ T ] .

To ensure that t a ​ has the same dimensions as velocity, a must have the dimension [ L ] [ T ] − 1 × [ T ] = [ L ] .



Second Term ( t 2 + c b t ​ ) :


For this term to have the same dimensions as velocity, both the numerator and the denominator must have compatible dimensions.

The numerator b t has the dimension b × t = b [ T ] .

The denominator t 2 + c should have dimension [ T 2 ] , where t 2 naturally has dimension [ T 2 ] . Therefore, c must have the dimension [ T 2 ] for t 2 + c to have the same dimension.

Hence, b should have the dimension [ L ] [ T ] − 3 , so that b t gives [ L ] [ T ] − 1 , matching the dimension of velocity.


Therefore, the dimensions are:

a : [ L ]

b : [ L ] [ T ] − 3

c : [ T 2 ]


These are the dimensions that make the equation dimensionally consistent.

Answered by MasonWilliamTurner | 2025-07-07