A projectile motion under air resistance

A projectile motion under air resistance

Physics

Introduction

We set the initial value at rest. Provide the velocity and the mass of the object. For the first case, we are using a grain of different mass under free fall. When the grain is released from a height of 5cm, air resistance acts on the grain. The time it takes for the object to reach the surface increases, whereas the velocity also increases.

The higher the velocity of the object, the higher the resistance. The graphs illustrate how air resistance acts on the objects under free fall. The initial velocity for the grain is zero(0), and it increases as air resistance works on the object.

The same applies to projected objects except that the velocity reduces due to the action of air resistance. The graphs show how air resistance acts on the project’s objects.

The ipynb files contain the python code. You can open them using jupyter notebook available online.

For exercise one, we have attached the function.ipnyb file. The graph has been plotted with the quantities bv and cv2, both as a function of x = Dv. Their relative magnitudes have been compared and established the range of values of Dv for which the linear term can be neglected and the range for which the quadratic term becomes negligible and the range for which both terms must be included. The visual studio editor can be used to access the codes since adding them in the report makes it look untidy.

2. f(v) = bv + cv2

b = BD

c = CD2

1.6×10−4 Ns/m2(0.07)(5m/s)+ 0.25 Ns2/m4 ( 0.07)(0.07) = 0.0001281

1.6×10−4 Ns/m2 (1.5×10−6 m) (5×10−5 m/s) + 0.25 Ns2/m4(1.5×10−6 m) ) (5×10−5 m/s) = 0.1875

To begin with we are requested to plot graphs for a falling grain from a height of 5cm. so we will begin with graphs showing how air resistance acts on the grain as its falls. We will try out different masses of grain from a range 0.1 to 1.0 in grams.

A graph for an oil drop

2. A projectile motion under air resistance

The following graphs show air resistance acts on a projected object. Aspherical object with an initial velocity which we have set in the fv function, forming an angle which we have also set the value.

In the fv function, the trajectory angle can be changed.

A graph for a trajectory

Determining the optimum launching angle for a trajectory

1. Dvt from the graph= 10m/s

Dvt from the formula = 9.80968

1. The code is attached to the file.
2. The graphs are as plotted above. Acceleration is inversely proportional to the mass of the object, and this means that more massive objects will fall faster than lighter once when thrown at the same speed and the same angle. In the presence of air resistance, more massive objects experience higher terminal velocity hence falls faster.
3.  Results obtained from the graph = 10m/s

Results obtained from the formula =4.4905(10ᶺ6)

1. As shown in the above graph objects fall at different speeds when they have different masses, the assumption that objects fall at the same rate if they have different masses is not true as shown in the experiment and the graph. Objects with heavier masses tend to fall faster than objects with lighter masses, meaning that acceleration is inversely proportional to mass. This approximation is as good as only when dealing with objects whose masses are almost identical, considering all the other variables are the same, e.g. shape, speed and angle of launch. Heavier objects reach terminal velocity faster than lighter objects that explain why heavier objects fall quicker.

3. for exercise 4: projectile motion under air resistance

Projectiles in a vacuum tend to go to higher terminal velocities and hence experience more significant displacement than objects moving through air. It is illustrated by a steep curve and lower maximum height compared to the one in a vacuum that’s assuming the objects are thrown at the same angle n with the same linear velocity. Air resistance reduces the velocity and the displacement of the projectile as it moves through the air.

Conclusions

When objects are moving through the air, they face a drag. Drag is the frictional force an object encounters as it moves through the atmosphere. Maximum resistance occurs when the weight of the object equals the resistance force.

Air resistance reduces both the velocity of the object as it moves through the air and the distance the projectile goes.

The maximum height reached by the projectile is also affected. The maximum height reduces when the object faces air resistance.

Air resistance is not equal in both direction and magnitude.

Drag force is directly proportional to the square of velocity FD ∝v2.

The frictional force is also directly proportional to the speed   FD=12ρv2CD

Air resistance acts in the opposite direction of the instantaneous motion of the object, and the magnitude is directly proportional to the object.

In the presence of air resistance, the object tends to fall more steeply than it rises. Indeed, in the presence of strong air resistance, the object almost falls vertically.

More massive objects reach terminal velocity faster than lighter ones, thus explaining why more massive objects fall more quickly.

Citation

http://www.schoolphysics.co.uk/age16-19/Mechanics/Kinematics/text/Projectiles_and_air_resistance/index.html

https://physics.stackexchange.com/questions/389590/why-does-air-resistance-affect-projectile-motion-in-the-way-it-does

https://farside.ph.utexas.edu/teaching/336k/Newtonhtml/node29.html

https://answersdrive.com/how-does-diameter-affect-the-motion-of-a-projectile-1243389

https://en.wikipedia.org/wiki/Projectile_motion