AP Physics C Mechanics Calculus Basics

Transcription

1 AP Physics C Mechanics Calculus Basics

2 Among other things, calculus involves studying analytic geometry (analyzing graphs). The above graph should be familiar to anyone who has studied elementary algebra. The horizontal axis is the 'X' axis and the vertical axis is the 'Y' axis. The primary concern of differential calculus is determining slopes of equations. Since this is a primer, we will review the concept of slope using a linear (straight line) equation. The slope of a line (designated by the letter 'm') is defined as the difference in 'y' divided by the difference in 'x'. (or the "rise over the run" as it is sometimes called). Mathemeticians use the Greek letter delta "D" to represent "difference" and so this equation could be written: Using y=3x + 6 (the red line in the graph above), we take the 2 points (x=2, y=12) and (x=-2, y=0) and calculate the slope: Although the equations are different, if we did slope calculations for the yellow or blue lines, they would have exactly the same slope as the red line (m=3).

3 There is an easy method to calculate slope. For equations of the form y = a x + b = 0, the slope equals 'a' (the coefficient of 'x'). This is better than choosing points and calculating differences. Just one more quick example - what is the slope of 5y = -3x + 7? Since the equation has to be of the form y = ax +b then y = -3/5 x + 7/5 and so the slope = -3/5.

4 Differential (what s that slope) Calculus In the previous section, we learned how to calculate the slope of a linear equation (equations whose exponents = 1). What happens when dealing with quadratic, cubic and higher-power equations? In other word a function that is not a straight line.. The graph at the right is based on a quadratic equation which predicts the distance an object has fallen (the y-axis) in relation to time (the x-axis). You cannot state one specific slope for this equation because the slope is constantly varying. What can be calculated is the slope at any point along the curve. Let's calculate the slope when time = 3 seconds. The formula states the distance (in feet) = ½ g t² (where g = 32 feet/sec²). So, when t = 3 seconds, d= ½ 32 3² = 16 9 = 144 feet. So, we have our first set of values: y=144 and x=3 But what do we do for choosing a second point? We could try using a value of t= 4 seconds, remembering this is not the same slope at t=3. So, we get d= ½ 32 4² = = 256 feet. So, we have our second set of values: y=256 and x=4 Calculating this approximate slope yields: Why don't we choose a closer value of x such as 3.1? When that is the case, the distance equals feet and our slope is: How about choosing a value of x that is even closer to 3 than 3.1? Besides representing a difference, Dx (called 'delta x') also represents the smallest possible quantity greater than zero. D x is less than a millionth, less than a trillionth - it's 1 divided by infinity. So, when x2 = 3 + D x, then y2 equals ½ g t² = 16 * (3 +Dx)² = 16 * (9 + 6 Dx + D x²) = D x + 16 Dx² and the slope at x = 3 can be calculated as: Since Dx is such an incredibly miniscule quantity, we can safely say that at x=3, slope = 96. Incidentally, the process of calculating a slope is called differentiation, the result of these calculations is called the derivative and this branch of mathematics is called differential calculus. The derivative is usually represented by dy/dx or f'(x).

5 Differential Calculus Although the previous method does work, it has 2 drawbacks - it is rather cumbersome and it only calculates the slope at one particular point. If we wanted to know the slope at x = 2, we would have to go through all those calculations again. Is there an easier way for determining the slope of an equation at any point? Yes Differentiation the Easy Way For a function of the form k x n, the derivative is equal to n k x (n-1) An example is the equation d= ½ g T² (or d = 16 T²) The derivative equals 2 16 T or 32 T Having determined the derivative, we can put it to use by the previous example when we calculated the slope for x=3. When x = 3 (or time = 3 seconds), the slope = 32 3 or 96. What about the slope at 2 seconds? 32 2 equals 64. The derivative of a constant (for example the number 7) is always zero. So, by way of example, the derivative of x2 + 7 is 2 x. Also, if an equation has more than 1 'x' term, simply differentiate each term and then sum those derivatives. Example: What is the derivative of 3x 3-5x 2 + 2x + 13? Answer: 9x 2-10x + 2 One important point to remember is that this method of differentiation works ONLY for equations of the form k x n.

6 Integral Calculus (What s under that curve!) As we learned, differential calculus involves calculating slopes and now we'll learn about integral calculus which involves calculating areas. The graph to the right where velocity = g T (or v = 32 T), is based on the derivative of the second graph equation d= ½ g t². Now, if we wanted to determine the distance an object has fallen, we calculate the "area under the curve". If we calculated the sum of the orange, blue and red areas this would equal the distance fallen after 3 seconds. Area of a Right Triangle = ½ (base * height) = ½ (3 seconds * 96 feet per second) = 144 feet Now looking at the previous graph, we see that this is the precise distance after 3 seconds. If we wanted to find the distance fallen between 2 and 3 seconds, we calculate ALL the area from 0 to 3 seconds (144 feet) and then subtract the distance from 0 to 2 seconds: ½ (2 seconds * 64 feet per second) = 64 feet So, the distance fallen between 2 and 3 seconds is = 80 feet. Looking at the previous graph and doing the subtraction, we see the numbers are the same. So how were we calculating these areas? We multiplied the y- axis (which is the quantity g T) by the x-axis (time in seconds or 'T') and we multiplied this by ½. So we calculated the area by the formula ½ (g T T) which equals ½ g T², and this is the precise formula which was used in the previous section!! The process of calculating area is called integration, the resultant formula is called the integral and this branch of mathematics is called integral calculus. (NOTE: the integral is sometimes called the anti-derivative)

7 Integral Calculus Integration the Easy Way For a function of the form k x n the integral equals k x (n+1) (n+1) Example 1: What is the integral of 5x 2 + 3x -7? Answer: (5x 3 )/3 + (3x 2 )/2-7x + c where 'c' is a constant - the derivative of a constant is zero and so when calculating an integral we have to allow for a constant. Example 2: Here's an interesting example of integral calculus. The area of a sphere is calculated by the formula 4 p r ²? What is the integral (anti-derivative) of this formula? Answer: (4 p r ³)/3 which happens to be the formula for the volume of a sphere! You now have a basic understanding of calculus (both differential and integral) This was a very quick, "barebones" explanation. However, you should have enough for AP Physics C