# Confused about energy

#### damo_clark

I have learnt that the kinetic energy of a moving object is equal to its mass multiplied by it's velocity squared, divided by 2.

If I takes off in my 2 million kg spaceship from rest to 1 million metres/second, I have calculated from this formula that I will need 1 million, million, million Joules of energy.

However, to achieve a velocity of 2 million metres/second, my 2 million kg spaceship will need 4 million, million, million Joules of Energy.

Is this correct?

This confuses me...

Asume that Harry and Burt both get into a space ship for a trip out into space. They are both given an instruction to accelerate the rocket to 1 million metres/second. At launch Harry goes to sleep and Burt takes off. As per his instruction Burt accelerats the ship to a velocity of 1 million metres /second. Burt then retires for some sleep. When Harry wakes up, he thinks the ship is at rest, when in actual fact it is travelling at a constant velocity of 1 million metres/second. Following his instructions he accelerates the ship to 1 million metres/second relative to his initaial velocty. If my thinking is correct the final velocity of the rocket ship is 2 million metres/second relative to the initial starting point.???

Now, for such a trip I calculate the Burt used 1 million,million, million Joules of energy. Since, Harry. had exactly the same instruction as Burt, then Harry used 1 million, million, million Joules of energy, too. Adding the two together, total energy used is just the sum of the energy that Burt used and the energy Harry used which is 2 million, million, milllion Joules.

However, how can this be since the ship is travelling at 2 million metres/second and has a Kinetic energy of 4 million, million million Joules.

What am I doing wrong here?

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#### russ_watters

Mentor
Typically, I spell "Burt", "Bert"....anyway...

Energy is the capacity to do work, so the kinetic and potential energy equations can be derived from Newton's equations of motion. Acceleration is constant with force and distance is a square function, so the faster you go, the faster you need to expend energy to keep up the acceleration.

You also have noted there that energy is a relative quantity, which depends on your frame of reference. This is easy enough to see in a car accident, where a head-on collision does more damage than a rear-end collision because the energy is relative to the closure speed of the cars, not the speed of the cars relative to the ground.

#### AJ Bentley

Harry doesn't know how fast he's travelling.
All he can do is to burn enough fuel to get the ship from rest to 1 million m/sec. That is 1 million, million, million Joules of energy.

Unbeknown to him he is already travelling at 1 million.
So the increase in speed is not 1 million but only enough for a total KE of 2 million, million, million Joules of energy. Which is root 2 million m/sec.

Instead of a final 2 M m/sec he is moving at root 2M m/sec.

#### hikaru1221

Great point, AJ Bentley! I'm just wondering, how does the speedometer of the spaceship look like?

Back to the problem. I think there is another way to understand it. First, the ship is at rest. Burt suddenly "jumps" out of the ship and uses some kind of remote control to accelerate the ship to 1Mm/s relatively to Burt (he can use some radar gun to measure the ship's speed). So now, Burt is at rest. Then Harry wakes up and does the same - "jumps out" and accelerates the ship to 1Mm/s relatively to him. So now, Harry is moving at 1Mm/s, and the ship is moving at 2Mm/s, while Burt is still at rest. Is this what you meant, damo_clark?

If so, then the explanation is already proposed by russ_watters. Kinetic energy and work are relative quantities. The work done on the ship in the phase of accelerating it from 1Mm/s to 2Mm/s in the reference frame of Burt is different from the work in the reference frame of Harry. The law of energy conservation, however, remains valid in both cases.

#### physi-man

Your chain of thought is correct but you are indeed going about it wrong. When you are calculating this you want to calculate how much energy you are going to input and then get your speed. This is where you are correct and incorrect. When adding additional energy you want to take into consideration the initial energy you already contain.
I'm going to use scientific notation to make this easier. To calculate your speed your formula will be:
Energy/(mass)(.5)=√Speed
So with your initial input of 1x10^18 we get this
(1x10^18)/(2,000,000)(.5)=√1x10^12
The root of 1x10^12 is 1 million which is your current speed. Now instead of running this same calculation and adding to get 2 million you should instead use your current kinetic energy, which is 1x10^18, and add that to the additional input of 1x10^18 which will yield 2x10^18. Now that you have your current kinetic and your additional input of energy, run the same calculation again.
(2x10^18)/(2,000,000)(.5)=√2x10^12
The root of 2x10^12 is 1,414,213.562. <-----This will be your current speed, not 2million m/s.
So remember you must add your current kinetic energy to the amount of energy you will be adding to get your speed.

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