Suppose an object starts from rest and attains a velocity $v_1$.Its kinetic energy (and also change in kinetic energy) is $(1/2) mv_1^2$.
Now suppose that the object starts with velocity $v_0$ and attains an extra velocity of $v_1$. Here, the change in kinetic energy is $(1/2)m((v_0 + v_1 )^2 - v_0 ^2)$ and not $(1/2) mv_1^2$.
However, momentum change in both the cases are equal. Why does the energy change depend on the initial energy/velocity/momentum it had? Mathematically, I understand that :
- kinetic energy is proportional to $v^2$ and hence we cannot calculate change in energy as $(1/2)m(\Delta v)^2$
- Work done is force times displacement and if the object starts at a higher speed, its displacement during the application of force will be higher, hence mathematics gives us a higher 'work done'
But I want to understand the physical significance.
Why does the same momentum change not mean same energy change?
Why does work done depend on the displacement of the object that was not caused entirely by the force but was due to the velocity it initially had?
(edit: extra displacement of $v_0\Delta t$, due not to the force F but due to the initial velocity $v_0$ it already had , in other words, the force that is supposed to be doing work is not the cause of the entire displacement, and yet we multiply it by the entire displacement 's'!! wouldn't it be more logical if the displacement was restricted to that caused by the force only? and this makes me question the definition of F.ds for work done!)
Another question that I would like to ask (a related question, or perhaps conceptually the same doubt just rephrased):
say a force of $F_1$ can cause a displacement of an object to the right. A force $F_2$ is capable of causing a leftward movement (consider constant forces for simplicity).
If I apply the forces separately, say the displacements are $d_1$ and $d_2$(consider the forces applied for the same time interval $\Delta t$).
If I apply them simultaneously, let the displacement be $d_3 < d_1$ and $d_2$
Work done by $F_1$ in case 1 is $F_1.d_1$
Work done by $F_1$ in case 2 is $F_1.d_3$ < $F_1.d_1$
Net work done on the object = $(F_1 - F_2).d_3$ and not $F_1.d_1$ - $F_2.d_2$
I am not being able to digest this, and feel like there is a loss of something somewhere... why is the work done by $F_1$ lesser in the second case just because of the existence of another force? shouldn't it be independent ? where did that difference in work done/energy go to?
To be more explicit, suppose $F_1$ acts for a time interval of $\Delta t$ and then after its work $F_2$ acts for the same time interval $\Delta t$. The momentum change would be $F_1 \Delta t -F_2 \Delta t = (F_1-F_2) \Delta t$ - same as what would happen if both forces acted simultaneously during the interval $\Delta t$. Yet, the work done (by each of the forces, and even the net work on the object) and hence the change in energy will be different.
Why so? What is happening to that difference in energy in the two cases... -same forces, same change in momentum, just a small difference in the experiment in terms of when the forces are applied...
(I would like to get an explanation of the physical interpretation, because the math has already been worked out and interpreting it is the problem)