This investigation will look at the effects of air resistance on falling objects, where the objects will have the same dimensions but different masses.

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Introduction

This investigation will look at the effects of air resistance on falling objects, where the objects will have the same dimensions but different masses. I will firstly model these situations to make predictions. This can then be experimented to obtain a set of results which we can use to test the models. The objects that we used were paper cake cups. We can change the mass by stacking these cups together, where stacking has no significant change on the overall profile as it drops.

Modelling Assumptions

As I am modelling this, there are assumptions that need to be made in order for the models to be more practical:

  • The value of g (the acceleration due to gravity) is constant (9.81 m/s). This is probably not very significant as the actual mass of the paper cups is small, and so the value of g will be quite constant.
  • The centre of mass remains the same. The assumption will not be very significant as the centre of mass would be quite constant.
  • The paper cups reach terminal velocity. This is important as if the paper cups do not reach terminal velocity, we could get inaccurate values of k.
  • The cups used are uniform with equal masses. Not very significant as masses of cups are very small variations in mass with other paper cups is very small. Also, the paper cups do have virtually the same surface area when stacked.
  • There are only two forces which act on the paper cups, which are air resistance and weight, with no external forces (i.e. no downwards forces). This is quite significant as it could make readings inaccurate if there were any downward forces.
  • The motion of the paper cups is one dimensional meaning that the cups will fall straight down, with no horizontal motion. Quite significant as there would be some sideways motion. But we assume this as there would be more complicated equations if not assumed.
  • Model 1

In the first model, assume air resistance to be directly proportional to the speed. If

Then

Where k is a constant

From Newton’s second law, we know that F=ma, where F is force, m is mass and a is acceleration (which is downwards).  

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As acceleration is the rate of change of speed, and that speed is the rate of change of distance (x):

     

Dividing the equation by m will give

We can get the auxiliary equation:

 

From this we can get the complimentary function:

TF:

Substituting these values back into

We get:

Putting this into the complimentary function ...

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