The velocity v​ of a particle at time t is given by v=at+b/t+c,where a,b and c are constants. The dimensions of a,b and c are respectively.

To determine the dimensions of the constants a, b, and c in the equation for velocity, we need to analyze the given equation: \( v = at + \frac{b}{t} + c \). In this equation, \( v \) represents velocity, which has specific dimensions that we can use as a reference point.

Understanding Velocity

Velocity is defined as the rate of change of displacement with respect to time. Its dimensional formula is given as:

[v] = [L][T]^{-1}

where [L] stands for length and [T] stands for time.

Breaking Down the Equation

Let's analyze each term on the right side of the equation separately to find the dimensions of a, b, and c.

• Term 1: at

The term \( at \) suggests that the dimension of \( a \) must balance with the dimension of time \( t \) to yield the dimensions of velocity. Thus:

[a][T] = [L][T]^{-1}

From this, we can isolate the dimensions of \( a \):

[a] = [L][T]^{-2}

• Term 2: b/t

Here, \( b \) is divided by time \( t \). So, for the term \( \frac{b}{t} \) to have the dimensions of velocity, we can set up the equation:

[b][T]^{-1} = [L][T]^{-1}

By rearranging, we find:

[b] = [L]

• Term 3: c

The constant \( c \) is added directly to the velocity, which means its dimensions must match those of velocity:

[c] = [L][T]^{-1}

Summary of Dimensions

In summary, we have:

• [a] = [L][T]^{-2}
• [b] = [L]
• [c] = [L][T]^{-1}

This dimensional analysis helps us understand how each term contributes to the overall expression for velocity. It also reinforces the idea that physical quantities are interconnected through their dimensions, providing a systematic way to verify equations in physics. If you have any more questions about this topic or anything related, feel free to ask!