I didn't try my best to solve it, but really I'm not so sure and never done such an exercise so I'd like to see how to solve those types of exercises.

A projectile is fired horizontally by a canon situated a platform at an height of 44 m. It is fired with an initial velocity of 244 m/s. Suppose the ground is perfectly plane and horizontal.

Using conservative energy laws find out :

1) The magnitude of the vertical component of the velocity of the projectile when it reaches the ground.

2)Do the same calculus in the case the projectile is falling off from 44 m.

My attempt : I know that the modulus of the velocity won't change in the 2 cases, but I'll have to show it.

My real attempt : I know that \(\displaystyle E=\frac{mv^2}{2}+mgz\) when the weight (I call it "\(\displaystyle P\)" in equations) is \(\displaystyle \vec{P}=-m \vec{g} z\). It means I use a z-axis instead of the commonly used y-axis.

Having said that, the horizontal and vertical components confuses me when I must use the law of conservation of energy. I feel very unsure.

I start saying that the energy of the system is the same at the beginning and at the final. Mathematically I wrote \(\displaystyle \frac{mv_i^2}{2}+mgz_i=\frac{mv_f^2}{2}+mgz_f\). The mass cancels out, so it's \(\displaystyle \Leftrightarrow\) to say \(\displaystyle \frac{v_i^2}{2}+gz_i=\frac{v_f^2}{2}+gz_f\).

I believe I'm looking for \(\displaystyle v_f\). Oh, I just got an image in my head. Once I get \(\displaystyle v_f\), I get the modulus of \(\displaystyle v_f\), right? It means that I just have to play with Pythagoras laws in order to get \(\displaystyle v_{f,z}\) because I already have \(\displaystyle v_x\) and because it is a constant I have \(\displaystyle v_{x,f}\). So let's try it out. \(\displaystyle v_f^2=v_i^2+2gz_i-2gz_f\) So \(\displaystyle v_f=\sqrt{v_i^2+2gz_i-2gz_f}\). As \(\displaystyle z_f=0\), it simplifies at \(\displaystyle v_f=\sqrt{v_i^2+2gz_i}\) Which is worth 61260.8 m/s. Hence \(\displaystyle v_f\) is approximately equal to 247.5091917 m/s. Which seems possible to me, but below what I expected. So Pythagoras says \(\displaystyle v_{z,f}\) is approximately equal to 41.53... well, doesn't make sens. What did I wrong?

A projectile is fired horizontally by a canon situated a platform at an height of 44 m. It is fired with an initial velocity of 244 m/s. Suppose the ground is perfectly plane and horizontal.

Using conservative energy laws find out :

1) The magnitude of the vertical component of the velocity of the projectile when it reaches the ground.

2)Do the same calculus in the case the projectile is falling off from 44 m.

My attempt : I know that the modulus of the velocity won't change in the 2 cases, but I'll have to show it.

My real attempt : I know that \(\displaystyle E=\frac{mv^2}{2}+mgz\) when the weight (I call it "\(\displaystyle P\)" in equations) is \(\displaystyle \vec{P}=-m \vec{g} z\). It means I use a z-axis instead of the commonly used y-axis.

Having said that, the horizontal and vertical components confuses me when I must use the law of conservation of energy. I feel very unsure.

I start saying that the energy of the system is the same at the beginning and at the final. Mathematically I wrote \(\displaystyle \frac{mv_i^2}{2}+mgz_i=\frac{mv_f^2}{2}+mgz_f\). The mass cancels out, so it's \(\displaystyle \Leftrightarrow\) to say \(\displaystyle \frac{v_i^2}{2}+gz_i=\frac{v_f^2}{2}+gz_f\).

I believe I'm looking for \(\displaystyle v_f\). Oh, I just got an image in my head. Once I get \(\displaystyle v_f\), I get the modulus of \(\displaystyle v_f\), right? It means that I just have to play with Pythagoras laws in order to get \(\displaystyle v_{f,z}\) because I already have \(\displaystyle v_x\) and because it is a constant I have \(\displaystyle v_{x,f}\). So let's try it out. \(\displaystyle v_f^2=v_i^2+2gz_i-2gz_f\) So \(\displaystyle v_f=\sqrt{v_i^2+2gz_i-2gz_f}\). As \(\displaystyle z_f=0\), it simplifies at \(\displaystyle v_f=\sqrt{v_i^2+2gz_i}\) Which is worth 61260.8 m/s. Hence \(\displaystyle v_f\) is approximately equal to 247.5091917 m/s. Which seems possible to me, but below what I expected. So Pythagoras says \(\displaystyle v_{z,f}\) is approximately equal to 41.53... well, doesn't make sens. What did I wrong?

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