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The motion of projectiles

Galileo also considered the motion of projectiles. He showed that their motion can be decomposed in a motion along a vertical and horizontal directions. Thus if a ball is thrown horizontally (and air friction is ignored) it will move in the horizontal direction with constant speed; in the vertical direction it will experience the pull of gravity and will undergo free fall. The use of this can be illustrated by the following situation. Suppose a ball is let fall from a height h and is found to take t seconds to reach the ground. Now suppose that the ball is instead thrown horizontally with speed v, what distance will it cover? The answer is v t because the ball, even though it is moving horizontally, in the vertical direction is still freely falling: the two motions are completely independent! (see Fig. 4.4). This, of course, was of great use in warfare.


Figure 4.4: Horizontal and vertical motion are independent: the cannon shoots the ball horizontally at the same time the hand drops its ball; they both hit the ground at the same time.  
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As another experiment consider the ``shoot the monkey'' demonstration (Fig. 4.5). The setup is the following: a hunter wants to shoot a monkey who is hanging from a branch. As soon as he shoots the monkey lets go of the branch (thinking that the hunter aimed at the branch, he believes that the bullet will miss him). But the bullet, to the monkey's surprise (and distress), does hit him! [*]


Figure 4.5: Shoot the ``monkey'': an illustration of motion in two dimensions.  
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The reason is the following: if there were no forces the bullet would go in a straight line (as indicated by the dotted line in the figure) and the monkey would not fall. So the bullet would hit the monkey. Now, since we have a force acting on the system (gravity) the monkey will not stay at rest but will accelerate downward. But precisely the same force acts on the bullet in precisely the same way, hence the bullet will not go in a straight line but will follow the curve indicated in the figure. The deviation from their force-free motions (rest for the monkey, straight line for the bullet) are produced by a force which generates the same acceleration in both objects, hence these deviations are precisely matched in such a way that the bullet hits the monkey.

Now, given a force of constant strength, it will affect bodies in varying degrees; the more massive the object the smaller the effect: a blow from a hammer will send a small ball flying, the same blow will hardly affect a planet. On the other hand gravity produces the same acceleration on the monkey and the bullet; that is why the monkey is hit. Since the mass of the monkey is very different from that of the bullet we conclude that gravity's force is very different for each of them. The fact that the accelerations are independent of the mass but the force is not is actually a very profound fact: the whole of general relativity is based on it (Chap. 6).

next up previous contents
Next: Astronomy Up: Mechanics Previous: The motion of falling

Jose Wudka