The ½ term comes from the formula for the area of a triangle. The triangle has a width of Δt, and a height of aΔt which we know from equation #2. The displacement due to acceleration is represented by the green triangle. The red rectangle is the contribution from the original velocity of the object. If we look at the area under the curve, we can break it into a rectangle and a triangle. Using geometry, we can examine the area under a curve of a velocity vs time graph for constant accelerating motion. If we assume that the original position and time are zero, we can further reduce this to This equation simplifies even further to become It is worthwhile to note what happens when the original velocity, v o, is zero. We simplify the remaining two terms to arrive at Next, we distribute the Δt term, and simplify by combining the v o terms. We then substitute in the definition for average velocity from equation #3.įrom here we subsitute in the final velocity arrived at in equation # 2 Using algebra, we can derive equation #4. The average velocity is the mean of the original and final velocity.įrom these three basic definitions we can derive the next two equations using either geometry or algebra (or calculus).
The connection between these is presented in the third equation which is simply the law of averages. It is important to notice that the first equation uses average velocity, whereas the second equation uses the change between the original and final velocity. The first two equations we have seen before.
In these equations, v is velocity, x is position, t is time, and a is acceleration. The last point in this passage he presented is that velocity increased with the square of the distance down the ramp.īuilding on what you have learned so far and what Galileo presented, we have what my physics teacher, Glenn Glazier, liked to call the Five Sacred Equations of Kinematics for constant acceleration. He presented that position increased with the square of the time, which is often referred to as the Law of Falling Bodies. Galileo determined this by having the rolling ball trigger bells as it rolled.īasically, Galileo presented that not only was the acceleration down a ramp due to gravity constant, but that the velocity increased linearly with time. As its velocity is increasing, the distance that it travels in each unit of time increases. The basis for the Law of Falling Bodies is that as the ball rolls down the ramp it accelerates. Advanced students may derive these same equations using calculus. Interestingly, Galileo’s proof used classical Euclidean geometry (which would be unfamiliar to the modern student of textbook geometry) instead of algebra, which is what we will present here. This is often known as the Law of Falling Bodies. Secondly, you were calculating in the Int variable which is ineffective in multiplication and division so instead, I used float which has better accuracy.In his Dialogues of Two New Sciences, Galileo derived the relationship between distance traveled and time as balls rolled down an inclined plane. In some versions, this doesn't give an error but here it does Ok so the problem was that you were taking the input in the form of an Int but you asked the user to enter '/' which is a string. Vi = int(input("What is the intial velocity?"))Ī = int(input("What is the acceleration?")) It seems my image isn't uploading so heres a gyazo link for it vf = input("What is the final velocity?") But if I remove the int() from each variable it wont let the equations work! I'm stuck, if anyone can give me some input it'd be greatly appreciated. But the problem is, when I try to enter the variable as / it gives me an error because I'm using integers. Now what I'm trying to do is make it so if you don't know the variable, you input / which is the variable you solve for. Hey guys what I'm trying to do here is make a very basic kinematics variable solver, so I've got the basic equation vf = vi + at.