1. Find the uniform acceleration

Since the body starts from rest and reaches speed $v_1$ in time $T$ under uniform acceleration,

$a = \frac{v_1 - 0}{T} = \frac{v_1}{T}$

2. Write velocity as a function of time

For motion starting from rest with constant acceleration,

$v(t) = at = \frac{v_1}{T}t$

3. Find instantaneous power

Instantaneous power is

$P = \vec F \cdot \vec v$

Since force and velocity are in the same direction,

$P = Fv = mav$

Substitute $a = \dfrac{v_1}{T}$ and $v = \dfrac{v_1}{T}t$:

$P = m\left(\frac{v_1}{T}\right)\left(\frac{v_1}{T}t\right)$

$P = \frac{mv_1^2 t}{T^2}$

4. Match with the options

The expression obtained is

$\boxed{\frac{mv_1^2 t}{T^2}}$

This matches Option B.

5. Check other options briefly

• A: $\dfrac{mv_1 t^2}{T}$ has wrong dimensions for power.
• C: $\dfrac{mv_1 t}{T}$ also has wrong dimensions.
• D: $\dfrac{mv_1^2 t}{T}$ differs by a factor of $T$ and is incorrect.

Therefore, the correct answer is B.