I need opinions on a Projectile Motion problem that I made up

[AlexJicu08]

Recently I started making physics problems and I made one that I really like, but I would like some feedback from other people (how difficult it is, how enjoyable it is to solve this problem, what I could improve about it, etc). Here is the problem:

A cannon is fixed at height H relative to the Earth in point A. The length of the cannon is l, and it can rotate around point O, making an angle φ with the horizontal axis.
A sphere-shaped projectile of radius R and density ρ leaves the cannon with speed v0 under the same angle φ. Friction with air has the value Fr = λρ0Sv^2, where S is the cross-section area of the sphere, v is its velocity, λ is a coefficient and ρ0 is the density of air. This force acts opposite to the velocity. The air also acts on the projectile with a buoyant force equal to the weight of the air displaced by the object.

a) Determine the SI unit for λ

b) Express the horizontal distance relative to point O at which the projectile gets as a function of l, H, φ, λ, ρ, ρ0, R, v0 and g

c) Calculate the distance d for φ ∈ {0°; 30°; 45°; 60°; 90°} (g=10m/s^2, H=2m, l=1m, v0=18km/h, ρ0=1.29kg/m^3, ρ=0.5g/cm^3, R=0.5m and λ = 1 (SI units)

d) Graph d as a function of φ evidentiating the values calculated at point c) and determine, from the graph, for what angle is d maximum

 

[wrobel]

Do you think you can solve the following easier problem?
A projectile of mass m is shot with initial velocity $\boldsymbol{\hat v}$ from the ground. At which distance from the cannon does the projectile fall?
$$m\boldsymbol{\dot v}=m\boldsymbol g-\gamma|\boldsymbol v|\boldsymbol v,\quad \gamma>0.$$

 

[AlexJicu08]

Do you think you can solve the following easier problem?
A projectile of mass m is shot with initial velocity $\boldsymbol{\hat v}$ from the ground. At which distance from the cannon does the projectile fall?
$$m\boldsymbol{\dot v}=m\boldsymbol g-\gamma|\boldsymbol v|\boldsymbol v,\quad \gamma>0.$$

What angle does the inițial velovity make with the horizontal axis?

 

[wrobel]

What angle does the inițial velovity make with the horizontal axis?

The vector initial velocity is given. Calculate.

 

[AlexJicu08]

Do you think you can solve the following easier problem?
A projectile of mass m is shot with initial velocity $\boldsymbol{\hat v}$ from the ground. At which distance from the cannon does the projectile fall?
$$m\boldsymbol{\dot v}=m\boldsymbol g-\gamma|\boldsymbol v|\boldsymbol v,\quad \gamma>0.$$

Also I never saw the last equation you write and I don't have any idea what it means

 

[AlexJicu08]

Also I never saw the last equation you write and I don't have any idea what it means

Does the dot mean derivative with respect to time?

 

[wrobel]

Also I never saw the last equation you write and I don't have any idea what it means

It is better if you would solve the problems from a textbook first and only after that compose them by yourself.

 

[PeroK]

Does the dot mean derivative with respect to time?

Feedback: your problem looks quite hard. Perhaps a numerical simulation might be the best approach.

 

[kuruman]

I think you should try solving the one-dimensional problem first just to see what is involved. Let the cannon shoot the projectile straight up, everything else remaining the same. That is a problem that can be solved analytically. You can calculate three relevant quantities

1. the total time of flight
2. the maximum height
3. the speed of the projectile when it returns to the point of projection.

Having done this, you will appreciate the complexity of what you proposed.

 

[AlexJicu08]

It is better if you would solve the problems from a textbook first and only after that compose them by yourself.

I'm still a student and i'm working with components of vectors (like vx and vy) and i mostly work with scalars, i don't know a lot about vectors and working with them. i could post my solution to the problem, if you want to see how i would work this out

 

[kuruman]

I'm still a student and i'm working with components of vectors (like vx and vy) and i mostly work with scalars, i don't know a lot about vectors and working with them. i could post my solution to the problem, if you want to see how i would work this out

Please do post your solution.

 

[AlexJicu08]

Please do post your solution.

I will write and I will post it by tomorrow