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Suppose we have a man traveling in an open car (roof open) with speed $v$ towards right (man faces right). He throws a stone (mass $m$) towards right, in his frame-forward with speed $V$.

In the car's frame, the total energy imparted to it was $$E=E_f-E_i=\frac{1}{2}mV^2$$
In the ground frame, the total energy given to it was $$E=E_f-E_i=\frac{1}{2}m((v+V)^2-v^2)=\frac{1}{2}m(V^2+2Vv)$$ What part am I getting wrong on? Is it okay to get such difference?

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    $\begingroup$ Hi Ashish. Don't get tensed that I tagged this as homework. This is a nice example of what our homework tag suggests. It is misused often. But, these questions still apply to our homework tag ;-) $\endgroup$
    – Waffle's Crazy Peanut
    Commented Mar 23, 2013 at 10:43
  • $\begingroup$ it was just to inform, i didn't mind anyway $\endgroup$
    – Ashish Gaurav
    Commented Mar 23, 2013 at 10:44

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You are not wrong. The only thing that may be wrong, is the interpretation. Energy is a conserved quantity, but only within an inertial reference frame. If you change to another reference frame, the energy will also be different. This is what you calculated.

Suppose you throw two rocks with the same velocity $V$. The first one, you throw at a target moving alongside the car. The second one you throw at a stagnant target by the side of the road. Which one will take more damage?

Brain teaser: Compare a car crashing at a parked car with velocity $V$ to a heads-on collision between to cars driving at velocity $\frac{1}{2}V$. Is this the same collision? (Extensive answers can be found elsewhere on this site)

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  • $\begingroup$ i think the stagnant target would be affected more, because then the impact would be more. If the car was with velocity $v$ and the stone had it as $q$ then the change in momentum would be from $m(v+q)$ to $0$. In the other case, only the throwing velocity causes the momentum change. Is this correct? $\endgroup$
    – Ashish Gaurav
    Commented Mar 23, 2013 at 10:50
  • $\begingroup$ The energies(my original question) are different in different frames. Agreed. Is the energy change(imparted energy) also different in different frames? $\endgroup$
    – Ashish Gaurav
    Commented Mar 23, 2013 at 10:53
  • $\begingroup$ Maybe you can join this chat-room: chat.stackexchange.com/rooms/8036/ashish-gaurav-and-bernhard To avoid lengthy discussion in the comments $\endgroup$
    – Bernhard
    Commented Mar 23, 2013 at 10:58

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