# Bow and Arrow problem

Let say we have one big massive bow that could accelerate a smaller bow instead of an arrow but this smaller bow could accelerate a small arrow. If the big bow accelerates the smaller bow at $$100$$ Km/H and we know that the small bow while standing still accelerates the small arrow at 100km/h will the combination of movements of both bows accelerate the arrow at $$200$$ Km/H.

• Is this different from, say, riding a motorcycle down the aisle of a moving train, or any other addition-of-velocity problem? – G. Smith Mar 27 at 17:40
• @JankoBradvica acceleration is not expressed in Km/h. You probably mean velocity. – Oбжорoв Mar 27 at 18:03
• The answer is yes - unless you want to start thinking of relativistic bows and arrows! – Mozibur Ullah Mar 27 at 18:07
• @MoziburUllah Your comment lead me to search google with phrase: 'archery arrow speed km/h' and got search result of 300 kmph.So do You think that the 'bows in series' can give the arrow a speed of 600kmph? – Janko Bradvica Mar 28 at 10:59
• @Oбжорoв I am saying just that for the arrow to gain the ending velocity it should be somehow accelerated by the bow..... – Janko Bradvica Mar 28 at 15:06

The non-relativistic velocity addition formula is

$$u=v_1+v_2$$

where $$u$$ is the final velocity as measured from a "stationary" observer A, $$v$$ is the velocity of the bigger bow measured from A and $$u$$ is the velocity of the smaller bow measured from the larger bow's frame. So $$u'=100\mathrm{km}/\mathrm{h}+100\mathrm{km}/\mathrm{h}=200\mathrm{km}/\mathrm{h}$$

Even if we were to consider relativistic effects and use the Einstein velocity addition formula

$$u=\frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}$$

the final velocity of the smaller arrow will be about $$199.999999\mathrm{km}/\mathrm{h}\simeq200\mathrm{km}/\mathrm{h}$$.

Of course the above calculations only hold true when we consider air resistance to be neglegible (which I would guess, is a pretty good approximation for an arrow).

First of all , I suppose you are talking about velocity . Now let's talk about your given problem. An arrow is shot with a velocity of 100 km/h relative to the smaller bow. Now the smaller bow also has a velocity of 100 km/h relative to bigger bow . so naturally the arrow has a velocity of 200 km/h relative to bigger bow . You can also use the concept of inertia instead of relativity to explain this