How can the kinetic energy equation be intuitively understood?
Momentum is proportional to an object's velocity, and kinetic energy is proportional to the square of its velocity
Looking at braking seems to help. If you're going twice as fast when you start to brake and your speed linearly goes to zero, it will take you twice as long at constant deceleration to decelerate to zero, and the average speed while braking will be twice as great (half your initial speed). This implies that the braking distance is proportional to the square of your initial speed, so you're applying a braking force for four times the distance. Similarly, accelerating a car to 40 MPH will take 4 times as much distance as accelerating to 20 MPH. But that doesn't quite get me to energy.
I think there's just one more step to making the relationship to kinetic energy obvious, but I don't have it.
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In short, momentum is vector and kinetic energy is scalar.
Two momenta in opposite direction is
If two objects move at the same speed then the more massive one has more quantity of motion and if two objects have same mass then the faster one has more quantity of motion.
Kinetic energy is energy of moving object.
An integral form might be more helpful to understand the difference,
Further reading :
- http://www.differencebetween.info/difference-between-kinetic-energy-and-momentum
- https://physics.stackexchange.com/q/16160/315444
Your analysis seems pretty good. To take an equivalent but more traditional example, imagine we throw a ball upwards. Ignoring air resistance and approximating the gravitational force as constant, the ball will accelerate downward with constant acceleration. If the initial (upwards) velocity is
Moving more towards your analysis, once we fix a constant force (and constant mass), then we know work is directly proportional to the distance,
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Change in momentum is force times time. On the other hand, change in kinetic energy is force times distance (more accurately, the component of the force along the movement times distance; a force perpendicular to the movement doesn't change the energy).
Now imagine you apply a force for a given time to an object initially at rest. The momentum will grow by the product of the force and the time. Since the object will have moved a certain distance after that time (and certainly in the direction of the force), its energy will also have increased by a certain amount. Also since a constant force means a constant acceleration, the velocity will also have increased by a certain amount.
Now imagine that you apply the same force for another second. For the momentum it's simple: The product of force and time is, of course, the same again, and therefore the momentum has grown by the same amount. Similarly, the acceleration was the same, and therefore the velocity has grown by the same amount. It is also easy to check that the same applies for a third acceleration phase, a fourth acceleration phase, and so on. Thus the growth of velocity and the growth of momentum are proportional.
On the other hand, if we look at the change of energy, we see that for the second acceleration, we get a greater energy growth, as the object was already moving, and therefore the total covered distance is larger.
So how much larger is the energy growth? Well, since the force was the same, it is the same as the difference of the covered distance. Now we had a constant force, and therefore uniform acceleration, which means the average velocity during the acceleration is just the arithmetic mean between the initial and final velocity.
So if we denote the final velocity of the first acceleration with
On the other hand, the average velocity of the second acceleration phase is
If we do a third acceleration phase, the average now is
This is related to the fact that the covered distance of an uniformly accelerated object is proportional to the square of the time of acceleration. Indeed, with constant force starting from rest, velocity is proportional to time, while covered distance is proportional to the square of time, thus covered distance (and therefore kinetic energy) is proportional to the square of velocity.
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