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

• AlexJicu08
AlexJicu08
Thread moved from the technical forums to the schoolwork forums
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

Last edited by a moderator:
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?
$$m\boldsymbol{\dot v}=m\boldsymbol g-\gamma|\boldsymbol v|\boldsymbol v,\quad \gamma>0.$$

wrobel said:
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?

AlexJicu08 said:
What angle does the inițial velovity make with the horizontal axis?
The vector initial velocity is given. Calculate.

wrobel said:
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

PeroK
AlexJicu08 said:
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?

AlexJicu08 said:
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.

SammyS and berkeman
AlexJicu08 said:
Does the dot mean derivative with respect to time?
Feedback: your problem looks quite hard. Perhaps a numerical simulation might be the best approach.

jbriggs444
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.

wrobel, PeroK and SammyS
wrobel said:
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

AlexJicu08 said:
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 said:
I will write and I will post it by tomorrow

## 1. What is the initial velocity of the projectile?

The initial velocity of the projectile is the speed at which it is launched. It is usually given in meters per second (m/s) and can be broken down into horizontal and vertical components if the launch angle is provided.

## 2. At what angle is the projectile launched?

The launch angle is the angle at which the projectile is fired relative to the horizontal. This angle significantly affects the range, height, and time of flight of the projectile. It is usually measured in degrees.

## 3. What is the maximum height reached by the projectile?

The maximum height is the highest point that the projectile reaches during its flight. It can be calculated using the initial vertical velocity and the acceleration due to gravity. The formula is $$H = \frac{(v_{0y})^2}{2g}$$, where $$v_{0y}$$ is the initial vertical velocity and $$g$$ is the acceleration due to gravity (9.8 m/s²).

## 4. How long does the projectile stay in the air?

The time of flight is the total duration the projectile remains in the air from launch to landing. It depends on the initial velocity and the launch angle. For a projectile launched and landing at the same height, the time of flight can be calculated using $$T = \frac{2v_{0y}}{g}$$.

## 5. What is the range of the projectile?

The range is the horizontal distance traveled by the projectile from the launch point to the landing point. It can be calculated using the initial velocity, launch angle, and the time of flight. The formula is $$R = v_{0x} \times T$$, where $$v_{0x}$$ is the initial horizontal velocity and $$T$$ is the time of flight.

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