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Question: A stick is thrown from a cliff 27 m high with an initial velocity of 18 m/s at an angle of 37 degrees above the horizontal. a) use the law of conservation of energy to determine the speed of the stick just before it hits the ground.

solution:

a busy cat http://i60.tinypic.com/2qmdz74.png

Just wondering why the vertical initial speed vector isn't used? Should 18sin37 be used? I think I am confusing myself with projectile motion.

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    $\begingroup$ Energy is a scalar not a vector, so direction isn't important. It is the magnitude of the velocity vector that is important for calculating energy. You don't have an energy in the x and y directions, you just have energy. $\endgroup$ – Kenshin Apr 27 '14 at 2:59
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The direction of the thrown object is not important since there is not air resistance. Whatever upward vertical component the object had will be downward when it passes you. Also, the horizontal component doesn't change. So it doesn't matter if you throw the object straight up, or at 37 degrees, or at 0 degrees, the object will always have the same energy when it is at the same level.

So the potential energy plus the kinetic energy at the start equals the kinetic energy at the end. $\frac {1}{2}m{v_i}^2 + mgy = \frac {1}{2}m{v_f}^2$

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The energy of the stick depends on its speed which is the magnitude of the velocity vector of the stick. Speed can be expressed as $v_1 = \sqrt{v_{1x}^2+v_{1y}^2}$, where $v_{1x}$ is the horizontal component of the stick's initial velocity vector, $\overrightarrow{v_1}$, and $v_{1y}$ is its vertical component. In this respect, you can see that the answer did in fact depend on the vertical component of the initial velocity vector. Part of the confusion arises from the answer's use of the symbol $v_1$ for speed. I'm sure the author of the question intended for the reader to infer that $v_1$ indicated a speed since it wasn't bolded or arrowed as in $\mathbf{v_1}$ or $\overrightarrow{v_1}$, indicating a vector.

Notice the usage of the words 'speed' and 'velocity' above. Velocity is a vector quantity that can be expressed either with a magnitude and a direction, or by the velocity vector's horizontal and vertical components. The word 'speed' is used to indicate the magnitude of a velocity vector. The question asked you to find the final speed of the stick, using its energy. The sticks energy can be determined using its speed and its height above ground. There was no need to express the speed of the object in terms of the horizontal and vertical components of the velocity of the object.

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As others have shown, whatever angle that we throw the stick with, as long as it has the same initial speed, it will hit the ground with the same energy.I just wanted to mention that different throwing angles will cause the stick to hit the ground in different directions. If the stick was a big heavy bowling ball and it was falling on a thin wooden surface, throwing with at 80 degrees causes the ball to hit the surface with a large vy and it will probably break it, While a ball thrown with the same speed and a smaller angle will hit the surface more horizontally and may leave the surface undamaged.

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