Units and Measurements

Let's restate the given problem carefully:
The velocity of a particle is given by
v=at+‌

b

t+c

We need to find the dimensions of a,b, and c.
Step 1. Dimensional formula of velocity
[v]=[LT−1]
Step 2. Dimensional analysis of first term at
In the expression v=at+‌

b

t+c

, both terms on the right-hand side must have the same dimensions as velocity.
The first term: at
‌[a][t]=[v]=[LT−1]
‌[a]=‌

[LT−1]

[T]

=[LT−2]
So,
[a]=[LT−2]
Step 3. The second term ‌

b

t+c

This term must also have the dimensions of velocity:
[‌

b

t+c

]=[LT−1]
Now, t+c is a sum - you can only add quantities with the same dimensions, so:
[c]=[t]=[T]
Then,
[b]=[LT−1]⋅[t+c]=[LT−1]⋅[T]=[L]
So,
[b]=[L]
Final Dimensions:
[a]=[LT−2],‌‌[b]=[L],‌‌[c]=[T]