Which of the following relations is correct? The symbols have there usual meanings.
P = power; d = distance; V = velocity a = acceleration.

Question

Which of the following relations is correct? The symbols have there usual meanings. 
P = power; d = distance; V = velocity a = acceleration.

Accepted Answer

The correct option is (3): $P=\vec{F}\cdot\vec{V}$

Power ($P$) is defined as the rate at which work is done or energy is transferred. In physics, work ($W$) done by a constant force ($\vec{F}$) over a displacement ($\vec{d}$) is given by $W = \vec{F} \cdot \vec{d}$. When this work is done over a time interval $t$, the average power is $P_{avg} = \frac{W}{t} = \frac{\vec{F} \cdot \vec{d}}{t}$. As time approaches zero, we consider instantaneous power. Since velocity ($\vec{V}$) is the rate of change of displacement with respect to time, i.e., $\vec{V} = \frac{d\vec{d}}{dt}$, we can express instantaneous power as $P = \frac{dW}{dt} = \frac{d(\vec{F} \cdot \vec{d})}{dt}$. If the force is constant or the instantaneous force is considered, then $P = \vec{F} \cdot \frac{d\vec{d}}{dt} = \vec{F} \cdot \vec{V}$. This relation is fundamentally important as it links the rate of energy transfer to the applied force and the velocity of the object. It's a scalar product because power is a scalar quantity, and both force and velocity are vectors.

The incorrect option is (1): $P=\vec{F}\cdot\vec{d}$
This relation represents work done ($W = \vec{F} \cdot \vec{d}$), not power. Work is the energy transferred when a force causes displacement, while power is the rate at which this energy is transferred.

The incorrect option is (2): $P=\vec{F}\cdot\vec{a}$
This relation is dimensionally incorrect for power. The product of force ($\vec{F}$) and acceleration ($\vec{a}$) does not yield units of power (Joules per second or Watts). Force has units of mass × length/time², and acceleration has units of length/time², so their dot product would result in mass × length²/time⁴, which is not power.

The incorrect option is (4): $P=\vec{V}\cdot\vec{d}$
This relation is also dimensionally incorrect for power. The product of velocity ($\vec{V}$) and distance ($\vec{d}$) would result in units of length²/time, which does not correspond to power. Power involves force, which is missing from this relation.