Is work force * distance or force * distance while the force was applied?

Question

Is Work

1. force times distance traveled by the object or

2. force times distance while the force was applied, without angles?

We see 1 everywhere but the distance only stops because of air resistance. This gives it no real purpose in physics.Work will be infinity if there is no energy conversion of kinetic energy because the distance will be infinity. This will make the energy transferred infinity which will violate the law of conservation of energy. So this cannot be true

2 is true since we need to find out how much energy is transferred. We can do that by multiplying the distance traveled while the force was applied and the force applied per distance. Since force is measure per distance we will find out the total energy transfered.

This means that Equation 2 is true while 1 isn't. Am I right or wrong. If i am right then why do we use equation 1

Answer 1

Work is energy transfer due to force times displacement in the direction of the force, or

$$W=\int \vec F\cdot d\vec s$$

Statement 1 is ambiguous since an object can continue to travel a distance after the force is removed, but work is only done while the force is applied..

Statement 2 is correct but poorly worded as may imply there can't be angle between the force and displacement. There can be an angle $\theta$ between the force and the displacement $s$ as long as $\theta\ne 90^o$. Then the work will be $Fs\cos\theta$.

Hope this helps.

Answer 2

2, assuming constant force directed in the same (or exact opposite) direction as the direction traveled by the object. A more general expression of the relationship between work and force requires basic vector algebra and a few semesters of calculus for path integrals (aka line integrals).