Kinetic energy's dependence on velocity.

[Yuqing]

This is not homework but rather a personal question.

Why is kinetic energy dependent on the square of the velocity. Is there a physical answer or is it simply mathematically derived. If there are some real life examples that can explain this then it would be greatly appreciated.

Similarly, why does it take more energy to accelerate an object at high velocities than low. For example, it takes more energy to raise an object's velocity from 100 - 200 m/s than it does to raise the velocity of the same object from 0 - 100 m/s.

 

[Astronuc]

This is the way I remember explained - using the product of force over distance, which becomes
$$W = m \int{v dv}$$, with an equivalence between work and energy.
http://scienceworld.wolfram.com/physics/KineticEnergy.html

So since kinetic energy is proportional to v₂, and the change would be related to v dv, which changes linearly with v.

Also, if one doubles v, e.g. v_f = 2 v_i, then E_f is proportional to v_f² = (2 v_i)² = 4 v_i² , or E_f = 4 E_i.

 

[nlightner]

The dependence of kinetic energy on velocity is not just a mathematical concept, but it has a physical explanation rooted in the laws of motion and conservation of energy. To understand this, we need to first define kinetic energy as the energy an object possesses due to its motion. The formula for kinetic energy is KE = 1/2 * m * v^2, where m is the mass of the object and v is its velocity.

Now, let's consider the motion of an object in a straight line. According to Newton's second law, the force acting on an object is equal to its mass multiplied by its acceleration (F = m * a). When we apply a force to an object, it accelerates and gains velocity. As the object gains velocity, its kinetic energy also increases. This is because the object now has more energy due to its increased motion.

The reason why kinetic energy is dependent on the square of velocity is because of the relationship between acceleration and velocity. When we accelerate an object, it gains a certain amount of velocity in a given amount of time. However, to double the velocity of the object, we need to apply the same force for a longer period of time. This is because as the object gains more velocity, it becomes harder to increase its velocity further. This is why the formula for kinetic energy has a squared term for velocity.

To answer the second part of your question, we need to understand that acceleration is the rate of change of velocity. In other words, it is the change in velocity over time. When we increase the velocity of an object from 0 to 100 m/s, we are essentially accelerating it. However, when we want to increase its velocity from 100 to 200 m/s, we need to accelerate it even further, which requires more energy. This is because as an object gains more velocity, it also gains more kinetic energy, and it becomes harder to increase its velocity even further.

In real life, we can see this principle at work in various situations. For example, when a car accelerates from 0 to 100 km/h, it requires a certain amount of energy. But to increase its speed from 100 to 200 km/h, it requires more energy as it has to overcome the resistance of air and friction. Similarly, a bullet fired from a gun has a much higher velocity than a thrown ball, and this is why it has much more kinetic energy.

In conclusion, the dependence of kinetic energy on