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Let's imagine that we fire a cannon located on a platform high above the ground and we want to compute the time after which the horizontal velocity of the cannonball will be zero. Let's assume that we shoot at 0 degree angle.

1. Air resistance force

[tex]F = C_d \frac{\rho v^2}{2}S[/tex]

2. Assuming that the force from point 1 is constant (terrible assumption but I hope it does not affect the answer too much) we have

[tex]F = \frac{mv}{t}[/tex]

[tex]C_d \rho v^2 \pi R^2=\frac{mv}{t}[/tex] assuming [tex]S=2\pi R^2[/tex]

[tex]t = \frac{m}{C_d \rho v \pi R^2}[/tex]

Now, let's make up some rational parameters

initial speed: 800 m/s

drag coefficient: 0.47

mass of the cannonball: 300 kg

Radius: 0.5 m

air density: 1.2 kg / m^3

Now, as I plug in my data to the equation, I get the answer - roughly 0.8 s. I know that this answer is not rational. My question is, is this significant error caused by the assumption that the force is constant?

1. Air resistance force

[tex]F = C_d \frac{\rho v^2}{2}S[/tex]

2. Assuming that the force from point 1 is constant (terrible assumption but I hope it does not affect the answer too much) we have

[tex]F = \frac{mv}{t}[/tex]

[tex]C_d \rho v^2 \pi R^2=\frac{mv}{t}[/tex] assuming [tex]S=2\pi R^2[/tex]

[tex]t = \frac{m}{C_d \rho v \pi R^2}[/tex]

Now, let's make up some rational parameters

initial speed: 800 m/s

drag coefficient: 0.47

mass of the cannonball: 300 kg

Radius: 0.5 m

air density: 1.2 kg / m^3

Now, as I plug in my data to the equation, I get the answer - roughly 0.8 s. I know that this answer is not rational. My question is, is this significant error caused by the assumption that the force is constant?

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