The Effect of the Angle of a Hill, Essay Example

The Effect Of The Angle Of A Hill On The Rate Of Change Of Velocity Of A Bike

The question revolves around the idea on how the angle that a hill makes with the ground affects the rate of change of velocity of a bicycle that is climbing it or moving down the hill or in other words, its acceleration as it climbs or slopes downwards. The SI unit of rate of change of velocity is meter per squared second (m/s²). Newton’s second law suggests that for an object to accelerate then it has to undergo the net result of any and all forces which act on it, and that is why, the law is summarized as Force = Mass × Acceleration, which further explains why the impact of a vehicle at a high velocity with a larger object is usually disastrous. The velocity decreases from that high point to zero within a very short time making the acceleration and so the force very high.

Acceleration can be shown by a scenario where a car is in three positions: when it is at a standstill which is said to be zero relative velocity when it is moving at an increasing speed and so-called acceleration and lastly when it is moving at a decreasing speed which is called deceleration. The formula for deceleration and acceleration is the same. An object that is moving at high speed or in other words is speeding up, or an object that is slowing down or in other words is decelerating, or an object that is changing direction is said to be accelerating hence their formula is the same. A bike rider will probably increase in velocity with time or accelerate faster when he is moving down the slope as the forces of gravity are pulling him down the slope. An object moving up a hill from the bottom will show a decrease in velocity with time or other words will decelerate due to the effect of forces of gravity on the bike rider which causes him to go slower and slower as he climbs the hill. The rate at which the bike rider decelerates while climbing the hill is dependent to a greater extent on the angle of the slope. The larger the angle the hill makes with the ground or the steeper the slope, the faster the rate of deceleration and so the larger the angle, the smaller the rate of change of velocity.

Acceleration is the second derivative of position concerning time, in math terms. The first derivative, of course, is velocity which can be explained further as follows: velocity refers to the rate of change of position whereas acceleration is the rate of change of velocity (Chien 2016). By definitions, velocity refers to the speed in a given direction while acceleration refers to the increase of rate or speed of something (N. 2013).

The average rate of change of any aspect can be solved with or without a graph, though graphs are preferred since they easily depict the information well.

The average rate of change = change in the y-axis/change in the x-axis; where the change in y refers to the vertical distance while the change in x-axis refers to the horizontal distance. For instance,

And in our case, that is in calculating the rate of change of velocity; the y-axis is represented by velocity while the x-axis denoted by time; hence the rate of change of velocity can, therefore, be calculated as; change in velocity over change in time. The gradeint of a velocity-time graph is usually the acceleration or the rate of change of velocity of the object in study whereby velocity unit is meter per second (m/s) while time’s unit is second (s), and hence the unit of acceleration can, therefore, be said to be meter per second squared (m/s²).

The whole calculation is done by drawing a tangent line touching on two parts of the curve which in our case will be the hill. The point stands for different angles along the slope of the hill.

In case the object is moving with an acceleration of +6 m/s/s, which means from the foregoing that it is changing its velocity by 6 m/s within a second, therefore, the gradient of the line will be +6 m/s/s and in case the object is traveling with a – 8m/s acceleration, then the slope would be – 8m/s/s which means it is decelerating or moving backwards. And lastly, 0m/s refers to the slope of the line is 0m/s.

Average acceleration is calculated in units of distance per square time (meters per second per second). It is the standard rate at which the velocity of an object changes within a specifiied time span which indicates the speed at which the object slows down, accelerated, or changes direction only. This equation therefore both  defines the average acceleration of a moving body and also proves the fact that it is the gradient of a velocity-time graph.  Just like velocity, acceleration is not constant if the graph is not a straight line which also implies that the moment the graph is a curve, for instance, the case of the hill, then at different points there will be a varied value of acceleration. The question, therefore, is to determine how the angle of a slope affects acceleration when one is cycling up the hill or down the hill.

Downslope Movement

Having introduced the basics, it is, therefore, appropriate to solve our problem. Whenever a bicycle rider goes down the hill, his/her velocity increases as the slope of the hill gets higher. To confirm the statement, we will evaluate the acceleration of the bike man down two ramps of different gradients to represent different angles of the hill and see how the slope affects acceleration.

There are general formulas that set the base for the argument:

Final = distance separating the end of ramp and finish line/ average time taken to travel from the end of the ramp to finish line

Acceleration = final velocity – initial velocity / average duration from release to bottom of ramp

Our proposition is that bike man would accelerate when moving downwards as it moves in the same direction of the acceleration and for the sake of our experiment, bike man would be substituted with marble and the hill would be substituted with a ramp.

Experiment 1: we will make a table like the one shown below

Set up the ramp by raising one side up, mark the initial point near the top of the ramp which can be even 30cm from the bottom, do the first trial by releasing the ball from the mark downwards and register the time it takes to get to the ground and record it in column A of the table. Repeat the same procedure and record as trial 2 in column A then get the average time taken from the same point.

Raise the ramp a bit higher and repeat the same procedure and record the results in column B plus their average.

Acceleration for ramp one which may be made an angle of 30⁰ = 0.3/3.39

= 0.08m/s²

Acceleration for ramp 2 which may have made an angle of 60⁰ with the ground = 0.3/1.8

= 0.166…m/s²

The above experiment proves that the steeper your slope/the larger the angle, the faster the marble will go downwards and this is shown clearly by the way acceleration for the second ram is higher than the first one. A steep slope/ a larger angle would affect the rate of acceleration down the slope by increasing it as the bike man would be much faster than with a gentle slope and the moment you are moving downwards then you will gradually increase your speed and that is why the marble took lesser time in ramp two than ramp one to reach the bottom. Hence, the larger the angle, the higher the rate of change of velocity down the slope.

Let’s focus may be on this second scenario:

Was the speed of the marble faster as it traveled further down the slope?

When a body left to fall freely, its velocity is increased by the force of gravity by 9.8 m/s within every second that passes. This is to say, after every second, the body would be falling freely at a 9.8 m/s velocity. In a similar way, after two seconds, the velocity of the free fall will have doubled and now the body would be falling freely at a 19.6 m/s velocity. Consequently, after three seconds the velocity triples and the body would be freely falling at a velocity of 29.4m/s. This shows that downslope movement will always result into an increase in acceleration (buddies, scientificamerican.com 2014). And from the above discussion, it was realized that the angle also influences the rate at which velocity changes in the sense that, the larger the angle the hill makes with the ground, the higher the rate of change of velocity down the slope and the smaller the angle it makes with the ground, the lower the rate of change of velocity.

Even though this experiment was not conducted in free drop (the availability of the ramp offers resistance to stop the marble), this same knowledge is relevant here. The marble should have continuously been accelerated by the force of gravity as it moved down the ramp in the experiment. This indicates that the longer the marble travelled, the faster it rolled. Any object placed along a tilted surface so that it rolls downwards will always slide down the surface, and the rate at which it slides below the surface is dependent upon the angle of tilt, how the surface is tilted, and it can be seen that the greater the tilt of the surface making a larger angle, the faster the rate at which the object will slide down and hence the higher the acceleration of the object.

Upslope Movement

From the above diagrams, it can be deduced that one has a smaller angle with the ground than the other and the problem therefore is, how they affect the rate of change of velocity while moving up the slope.

First, the bike man would take more time climbing the hill with the larger angle due to more friction, hilly parts and gravitational force which works on him. It indicates that the larger the angle, the lower the rate of change of velocity as will be shown below with the following example.

A cable is pulling a 200 kg block up an inclined plane elevated at 30 degrees and 60 degrees with 3.5KN of force. How fast is the block accelerating at the two degrees?

From the above calculations, it can be plainly seen that the larger the angle the hill makes with the ground, the less the acceleration and thus low rate of change of velocity which is observed by the fact that the angle 60⁰ after all the calculations has an acceleration of -0.28 m / s² while the angle at which the slope is at 30⁰ with the ground produces an acceleration of 9.1635m / s².

Alternatively, considering upslope movement where the bike man is at different points based on varied angles, we can also reduce the impact of angles on the rate of change of change velocity. For instance, at two points where the curve makes 20⁰ and 40⁰ with the ground for the first tangent, the velocity would be 80 m /s and 60 m/s respectively whereby 80 m/s is the initial velocity while 60 m/s being the final velocity for the first tangent. Assuming that the time taken between the two points is 40 seconds, we can calculate the rate of change of velocity (acceleration) which is the slope of the curve of the bike man using the equation below:

We can also calculate the acceleration at a much steeper part of the slope, for instance, two points where the hill makes 60⁰ and 80⁰ with the assumption that the velocity at the two points as 40 m/s and 20 m/s respectively where 40 m/s is the initial velocity while 20 m/s being the final velocity. Assuming also that the time taken between the two points is 60 seconds, acceleration can be evaluated using the method:

It can, therefore, be concluded that the angle a slope makes with the ground will always affect its rate of change of velocity for any moving body either down the slope or up the slope. For the bodies moving up the slope, the rate of change of velocity decreases as one moves up the slope where the angle gets larger as you climb upwards, consequently, bodies moving down the slope will always experience an increasing rate of change of velocity as the angle of the slope gets smaller.

Bibliography

Martin, James C., Douglas L. Milliken, John E. Cobb, Kevin L. McFadden, and Andrew R. Coggan. “Validation of a mathematical model for road cycling power.” Journal of   applied biomechanics 14, no. 3 (1998): 276-291.

Flint, J. J. “Stream gradient as a function of order, magnitude, and discharge.” Water Resources  Research 10, no. 5 (1974): 969-973.