Speeding up energy costs. Am I dumb?

Question

I have maybe very easy question but my friend keep telling me I am stupid.

Let's have 10kg mass in vacuum without gravity forces. To speed up this mass to 10m/s I need 500J.

Now I want to speed up this mass from 10m/s to 20m/s. It needs surely more energy for same amount of speed up right? I am again speeding up just 10m/s but I need 1500J now right?

My friend keep telling me its not true and it needs same amount of energy to speed up mass from 0-10m/s as when for example 1000-1010m/s

Where is the truth?

Answer 1

You are correct.

You put energy into something by doing work on it. You do work by applying a force through a distance.

This works no matter how you apply the force, but the math is easier to do when the force is constant, so let's assume it is. Then the acceleration of the object is uniform, that is, the speed initially increases linearly from zero to $10\ \mathrm{m/s}$. Suppose it takes exactly one second to reach that speed; it takes a total force of $100\ \mathrm{N}$ in the direction of travel to do this.

In that first second of motion, the object's average speed is $5\ \mathrm{m/s}$. So it travels $5\ \mathrm{m}$ in that time. The work done by a force of $100\ \mathrm{N}$ applied through a distance of $5\ \mathrm{m}$ is $100\ \mathrm{N} \times 5\ \mathrm{m} = 500\ \mathrm{J}.$

Now keep applying that same force for one second longer. The object speeds up from $10\ \mathrm{m/s}$ to $20\ \mathrm{m/s}$ during that time. The average speed is $15\ \mathrm{m/s}$ and the distance traveled during that second is $15\ \mathrm{m/s}$. The work done by a force of $100\ \mathrm{N}$ applied through a distance of $15\ \mathrm{m}$ is $100\ \mathrm{N} \times 15\ \mathrm{m} = 1500\ \mathrm{J}.$

There are a lot of things that are proportional when a constant total force is applied to an object moving in a straight line. Speed is proportional to the time since the speed was zero. Work done (and therefore energy put into the object) is proportional to the distance traveled. But distance traveled is not proportional to time, and therefore the things that are proportional to distance (such as energy) are not proportional to the things that are proportional to time (such as speed).

A non-constant force is more complicated but it still turns out that the kinetic energy of the object is still proportional to the square of its speed, not simply proportional to speed.

Answer 2

If you apply a constant force to the mass, its speed will increase linearly, and so does the power (force times speed).

The work increases quadratically because the space travelled increases quadratically. And accordingly, the kinetic energy increases quadratically. If the body is in free fall, this energy is drawn from the loss of potential energy, proportional to the space, also quadratic.