Kinematics equation and Work-energy theorem

I am actually amazed by how the work-energy theorem is related to one of the kinematics equations for constant acceleration motion.

The work-energy theorem states: the net work done on a particle is the change in its kinetic energy.

Mathematically it looks like this:

For the left-hand side of the equation, the net work done on a particle is the product of the net force and the displacement in the direction of the net force. So, we can write

I have used Newton’s 2nd law, which states that the net force on the particle is the product of mass and acceleration, in the above equation.

As for the right-hand side, the change in kinetic energy can be written as

Put all the ingredients back to the work-energy theorem, we get

The term m gets canceled out, and after rearranging, we get

Voila, isn’t that neat? The final equation is one of the kinematics equations for constant acceleration motion. We use this equation whenever the information about time duration is not given in the question.

What does this mean to me?

I am not sure who came up with the definition of work done. However, this example shows how interconnected the ideas in kinematics, force, and work are. They are not separate concepts but are different quantities that one can calculate based on the given information in the questions.

For example, suppose the question wants us to find the final velocity of the object after traveling some displacement under constant acceleration/force.

If the mass of the object is not given but the acceleration is given, then one can only use the kinematics equation to solve for the final velocity.

If the mass of the object and the net force acting on it are given, then you have a choice. Either by using the kinematics equation or the work-energy theorem. Both should give the same answer.

Note that this works for 1D motion all the time. That’s it for my sharing today.