Prediction of the relationship between the height used and the change horizontal distance travelled by the spherical mass.
I believe that height is directly proportional to distance squared. This means that if the height is doubled the distance will quadruple.
Explanation of the prediction in scientific terms.
The formula to work out distance travelled is simply = average speed/total time, however there are many other formula used in working these two out. First the speed, or velocity must be obtained. The velocity of the ball does not depend on the length of the ramp (if friction is discounted) but on the vertical height the ramp is set at or ‘h’.
Using previously obtained scientific knowledge I understand that GPE (gravitational potential energy) = kinetic energy. Therefore mgh (mass ï¿½ gravity ï¿½ height of ramp)=0.5mv2(0.5 ï¿½ mass ï¿½velocity squared). Due to there being mass on both sides of the equation it is mathematically acceptable to divide the mass out so the formula looks like so: gh = 0.
5v2. Using simple mathematical techniques we can rearrange that formula to provide us with the following equation which gives us the final velocity:
Velocity = V
Now it is possible to work out the final velocity of the ball as it hits the bottom of the ramp using this equation.
The second half of the formula to work out distance requires the time taken. Using required knowledge I know that it takes a same amount of time for a ball to drop from a certain height vertically, that it would if it was travelling with velocity horizontally.
To work out the time taken, for a ball to drop vertically downwards, one of the UVAST equations needs to be used. In particular the one relating to time which is: s (distance travelled) = [u (initial velocity) ï¿½ t (time)] + [0.5a (acceleration) ï¿½ t (time)]. As the ball is being dropped its initial velocity = 0. If anything is multiplied by zero the answer will be zero. It is therefore using appropriate mathematical we simplify the formula to: s = 0.5at2. Using once again simple mathematical procedure this formula may be rearranged as follows:
Time = V2s / a
It is now possible, knowing just the height of the ramp, the height that the ball will fall and gravity to work out how far the ball will travel. Distance = average speed ï¿½ total time taken. To determine the average speed we take the final velocity and divide it by 2 therefore:
Distance = 0.5V ï¿½ t
However, this does not fully explain why the height is directly proportional to the distance squared. The time taken for the ball to drop is always constant, the distance the ball is dropping will remain the same throughout and so will the acceleration (9.8ms-2), therefore we can disregard the time in dealing with the relationship. When working out the velocity the 2g is always the same, therefore that can be disregarded. Therefore the only variable is the square root of the height and so using mathematical deduction the distance must be directly proportional to the square root of the height. This can be rearranged as follows:
Distance2 ? Height
Brief account, including any data obtained, of the preliminary practical work carried out.
The preliminary work was used to provide good understanding in the practical and help us decide important parts of out method. Many apparatus set-ups were experimented with, ranging from the completely unfeasible to some that provided a large part of the final method. Once a suitable apparatus (see diagram below) had been achieved it was then important to judge the range of readings that were to be used. As a measurable minimum it was decided upon that a height of 10cm was suitable and therefore the lowest reading will be 10cm. Deciding the highest readings was much more difficult. Firstly, a height of 50cm was pursued as a maximum height however, the ball when leaving the ramp and reaching the horizontal was judged to be bouncing, this would have provided us with inaccurate readings. The next height chosen was 45cm and this was judged to be large enough to provide a good range of readings but small enough to prevent the ball from bouncing. The readings were made using simple techniques, where the ball hit the ground a member remembered the place and quickly measured it. The following readings were achieved:
The main thing I have decided is the range of my results. I will have results from 10 cm height to 45cm heights at 5cm intervals. However, I have also determined my method of measuring the distance travelled to be unreliable. Therefore I have devised a new way, using carbon paper. The carbon paper is placed covering the ground on and around the area where the ball might land. When the ball lands upon this paper a small mark is left this makes is much more accurate and reliable when measuring the distance travelled. As we have determined the mass is irrelevant however, too heavy a mass could be a possible safety hazard and therefore a relatively small marble sized metallic ball was used. the length of the ramp used may effect the readings as they provide friction so the same 1m length of ramp will always be used.
Variables, which must be kept constant to insure a fair test.
There are a few variables which must be kept constant to ensure a fair test and make my results as accurate as possible. The velocity with which the ball is rolled down the ramp must initially be 0 as this would effect the results, this is done by effectively letting go of the ball instead of pushing it. Another variable that must be kept constant is the direction that the ball must fly in. It must be straight for if it was not this deviance would greatly affect the reliability of my results. Also the ball must fly of in a completely horizontal direction for every reading.
Method (i.e. detailed experimental plan of what will be done in order to produce reliable and accurate data)
My apparatus will be set up as shown in the diagram. The clamp stand is fixed at the required height; this is measured using a meter rule. Checks will be made to make sure that the ramp is securely attached to the clamp stand and everything is suitably safe. A practice run will be made to make sure that the ball is flying horizontally and is not deviating at all, if this is the case the apparatus are changed accordingly. Sheets of carbon paper will be laid out near the launch ramp. The ball is then let go from the defined height and is allowed to continue its course until it reaches the ground. The mark on the carbon paper should be clearly visible and quickly yet precisely measured for the distance travelled. This experiment will then be repeated for all of my ranges until multiple readings are available for all required heights.
Explain clearly how your results would be processed and what your expectations are.
From the above method an accurate set of results ought to be produced and there are many things that can be done to these statistics that may be helpful. I suggest a scatter graph of height (on the x axis) against distance squared. I expect there to be a smooth gentle curve upwards, with possibly an anomaly on the way. An example graph that I have constructed is on the next page, this is how I believe my plan will lead to my results being like.
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