Kinetic Energy: Difference in energy for observers Consider a stationary 10kg bike; if I apply a force of 10J then $KE=\frac{1}{2}mv^2$ so $v=\sqrt\frac{2KE}{m}=1m/s$. If we consider the same example again, but from the point of view of a stationary astronaut looking down from space, the Earth is moving at approximately 30,000m/s, so the KE of the stationary bike would be $KE=\frac{1}{2}10*30,000^2=4,500,000,000J$. If we assume the bike is being pushed in the same direction the Earth is travelling, to get to 1m/s the bike will need a kinetic energy of $KE=\frac{1}{2}10*30,001^2 = 4,500,300,005J$, which is 300,005J more than for an observer on Earth. I understand that conservation of kinetic energy only makes sense if you look at all objects from the same frame, so my question is why it would appear to take so much more energy to accelerate a bike by 1m/s from one observer's point of view than another?
 A: 
I understand that conservation of kinetic energy only makes sense if you look at all objects from the same frame, so my question is why it would appear to take so much more energy to accelerate a bike by 1m/s from one observer's point of view than another?

All conservation laws are applicable  in a  closed systems.
A closed system is any physical system, contraptions for which all outside influences on the system are absent or negligible.
Though, it is difficult to devise an ideally closed system  as  no system can be perfectly closed. 
But if one  can measure  all outside influences precisely, we can make corrections for outside influences and  check  the conservation laws.
An automobile of mass m  moving at speed v on  a road has K.E. = (1/2)mv^2. 
But with respect to a train moving at the same speed parallel to the road, the automobile's kinetic energy is zero, for its relative velocity is zero with respect to the train. 
So the kinetic energy depends upon the measurement frame of reference.
But whatever inertial  reference frame one  uses, changes of kinetic energy will be unaffected by this choice.
Therefore if  one observes the change in Kinetic Energy of the Bike taking the Bike and its immediate environment as the closed system the necessary amount of energy required to provide the Bike a particular speed will be same  by the inertial observers.
If the  system is enlarged to add (moving earth + Bike) ; 
then also the change in energy of the bike will be effected by the same amount of work done.
As the energy of earth was already there when the astronaut looked at the  Bike on earth- so that amount of work/energy  may be adjusted (being common to both initial and final energy calculation).
