Terminal Velocity of a Cone

Introduction –

A falling paper cone reaches a steady speed called its terminal velocity. Terminal velocity is the maximum, constant velocity reaches by an object falling through a gas or liquid. I will be investigating what effect changing something about the cone has on the terminal velocity.

Planning -

The variable I am going to change each time in the experiment is the size of the cone. I will do this by varying the slant height on each of my cones.

I have written my method in a series of numbered points –

Make 7 cones of slant heights 3-10cm.
Measure 2m up the wall and mark - this is where the timing begins.
Measure another 50cm up the wall – this is where the cone will be dropped from each time, but it is not as important, as long as the cone reaches its terminal velocity before the 2m point.
Drop each cone from the designated point.
Time the cone from the 2m point until it hits the floor using a stop clock.
Repeat each measurement with the stop clock 3 times.

The equipment needed is –

Stop clock
Metre rulers
Paper
Sticky tape
Blu-tack

As the cone falls, the air resistance force – frictional force – increases as the speed of the cone rises. Eventually, the air resistance force, which limits the speed at which objects can move through the air, is enough to balance the weight of the cone. The cone then stops accelerating and falls with a maximum, constant velocity, called its terminal velocity. The cone accelerates until the driving forces (weight force or gravity) are balanced by the counter forces (air resistance) and it then reaches a terminal velocity.

Cones with bigger slant heights have larger surface areas. This therefore leads to an increased air resistance force pushing upwards. At terminal velocity, there is no resultant force, and so no further acceleration can occur.

Prediction -

Based on this scientific knowledge, I predict that cones that have larger surface areas (bigger slant heights) will fall to the ground slower, and therefore have a larger terminal velocity, than cones which have smaller surface areas. I therefore think that the cone with the smallest slant height in my experiment – 3cm – will have a higher terminal velocity than that of the largest cone in my experiment with a slant height of 10cm.

Fair test –

There are certain variables that need to be kept constant to make this a fair investigation. Each cone should be made in the same way – a circle drawn which a specific radius (then to become slant height), and then a 90° drawn on to mark the overlap, and the a line to be cut along the radius to the centre of the circle, and the circle made into a cone, and secured with sticky tape. Each cone should be made out of the same type of material – paper – and if possible the sticky tape ...

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Fair test –

I plan to have a cone with the slant height of 3cm as my smallest cone, and a cone of the slant height 10cm to be my largest cone. This therefore means that the size of my cones will vary from 3cm to 10cm, meaning that they are of the sizes 3cm, 4cm, 5cm, 6cm, 7cm, 8cm, 9cm and 10cm. This will give a good range of results and will make the analysis at the end easier.

I plan to make my results as accurate as possible by using a stop watch rather than a watch or normal wall clock. I plan to get another individual to stand on eye level with the point which the timings begin – 2m – so that they can see exactly when the cone passes this point. I will repeat all of the timings taken 3 or 4 times to identify any errors in my results.

I think that my prediction is sensible because the same theory, that the greater the surface area, the slower the speed is employed in such things as parachuting, and hang gliding and even sky diving. I will talk about parachuting, which heavily relies on the scientific fact that air resistance pushes up on the parachute and lowers the speed that the parachutists is travelling at, breaking the fall, and bringing them down at a steady speed – their terminal velocity. Parachutist rely on the fact that air resistance will push up on the large surface area of their parachute, for them it is a matter of life or death in most cases.

I made various sizes of cones and tried dropping them at the same time and simply seeing which one hit the ground first. I tried different heights, as dropping them from a sitting position did not show much of a difference in the times which it took them to hit the ground. But the higher I dropped them, the more difference I saw in the times which it took for them to hit the floor. This helped me to decide on a height from which to drop my cones in my experiment, to ensure that they did reach their terminal velocity at a reasonable distance before hitting the ground.

In previous lessons, a falling feather has been used to demonstrate terminal velocity. The feather accelerates due to its weight. Air resistance on the feather increases quickly and soon matches the weight force. The feather is then said to be at a terminal velocity. This again helped me in the planning of my experiment, by showing me when an object is at its terminal velocity.

Results -

Formula: terminal velocity = distance

time

Analysis –

My results show that as the size of the cone increased the terminal velocity decreased. The first graph plotted of terminal velocity against slant height was a curve so I plotted a second graph of the inverse of one of the quantities. The second graph shows the inverse terminal velocity (1/terminal velocity) against slant height of cone, and as this is a straight line it, shows that there is a linear relationship between the two variables, ie that terminal velocity is related linearly to size of cone.

The results show that when a cone has a small slant height, it moves faster through the air, because it experiences less air resistance force on its small surface area. The smaller the cone dropped, the higher the terminal velocity was, and conversely the greater the slant height, and therefore the greater the surface area, the more air resistance is experienced, slowing the rate of fall of the cone. This shows that as the slant height is increased, the terminal velocity decreases, the two variables have a linear relationship to one another.

The results have shown that my prediction that cones with a greater slant height will have a lower terminal velocity than cones with a small slant height was correct. My results produced a straight line graph showing that the two variables had a linear relationship to one another and I a m very confident that more readings would follow this pattern.

Evaluation –

As my results produced a line graph with most plotted points on or very close to my line of best fit, I believe that the quality of my evidence is good, and that I have proved my prediction to be correct.

The hardest part of this experiment to do accurately was the timings. This was because it was very difficult to start the stop clock at exactly the point it crossed the 2m mark on the wall. I would suggest that may be the 2m mark could have been made more visible and easy to see, maybe by having a marker off the wall. An automatic timing device would increase the accuracy dramatically. It would work by starting a timer when the cone went through a beam, and stopping the timer when it hit a pad of sorts at the bottom on the floor. That way there would be less human error in the experiment, but this sort of device would be expensive to buy, and not practical for everyone to use at a school.

The values of my repeated readings were close together, only tenth of a second difference for most of them. I think that this shows the reliability of my results and evidence to be good. I think that my evidence is of a good quality – when graphed it shows a pattern, which I think, will carry on through different sizes of cones – and so I am sure of my conclusion.

There are two other experiments that I would like to suggest to provide extra evidence to support the results I already have.

My first experiment would be to make mini parachutes of varying size, using cloth or polythene. They would be weighted with a fixed load, the same for each. The parachutes would then be dropped, and timed over the same distance as the cones. The parachutes would provide a large surface area, which would increase, as the size of the parachute got larger. I would predict that as the parachutes got bigger, the terminal velocity of the parachutes would get smaller like the cones, as this experimental prediction is based on the same principles as that of the cone.
The second experiment would be to use cylinders of differing sizes. There would be one open end, and one closed end. They would be made out of paper, and have differing radii. They would once again be dropped, and timed over the same distance as the cones using a stop clock. It does not really matter the distance over which the cones or mini parachutes are timed, as long as they are at their terminal velocity during the fall. I decided to keep the distance the same as the cone to make the analysis easier, and I know that a shorter distance makes accuracy in timing harder using a stop clock. I would expect the same pattern to occur once again in the cylinders – of the two variables having a linear relationship to one another – because once again the larger the radius of the cylinder, the larger the surface area it will have.

Alison Hines 4L