Scalise said:Hello,
Someone could explain me why in the derivation below the mass m is divided by 2 in the last step?:
##\int\vec{F}\cdot d\vec{s}=m\int\frac{d\vec{v}}{dt}\cdot\vec{v}dt=\frac{m}{2}\int \frac{d}{dt}(v^{2})dt##
The work-energy theorem is a fundamental principle in physics that states that the work done on an object is equal to the change in its kinetic energy. In other words, when a force is applied to an object and it moves, the work done by that force is equal to the change in the object's velocity.
The equation for the work-energy theorem is W = ΔKE = ½mv2f - ½mv2i, where W is the work done on the object, ΔKE is the change in kinetic energy, m is the mass of the object, vf is the final velocity, and vi is the initial velocity.
The work-energy theorem is closely related to the principle of conservation of energy. This principle states that energy cannot be created or destroyed, only transferred or transformed. In the context of the work-energy theorem, the work done on an object by a force is equal to the change in its kinetic energy, which is a form of energy. This demonstrates the conservation of energy as the work done is equal to the change in kinetic energy, meaning the total energy of the system remains constant.
One example of the work-energy theorem in action is a rollercoaster. As the rollercoaster car travels along the track, it gains kinetic energy due to the work done by the force of gravity. When it reaches the bottom of a hill, its kinetic energy is at its maximum, and as it climbs up the next hill, its kinetic energy decreases as the work done by gravity is negative. Another example is a person throwing a ball. The work done by the person's arm causes the ball to gain kinetic energy as it moves through the air.
To apply the work-energy theorem to solve problems, you must first identify the forces acting on the object, the distance it moves, and the change in its velocity. You can then use the work-energy theorem equation to calculate the work done on the object and the resulting change in kinetic energy. This can be useful in a variety of scenarios, such as calculating the force needed to move an object a certain distance or determining the speed of an object after being acted upon by a force.