How to Define Acceleration

How to Define Acceleration Acceleration is the rate of change of velocity as a function of time. It is a vector, meaning that it has both magnitude and direction. It is measured in meters per second squared or meters per second (the objects speed or velocity) per second. In calculus terms, acceleration is the second derivative of position concerning time or, alternately, the first derivative of the velocity concerning time. Acceleration- Change in Speed The everyday experience of acceleration is in a vehicle. You step on the accelerator, and the car speeds up as increasing force is applied to the drive train by the engine. But deceleration is also acceleration - the velocity is changing. If you take your foot off the accelerator, the force decreases and velocity is reduced over time. Acceleration, as heard in ads, follows the rule of the change of speed (miles per hour) over time, such as from zero to 60 miles per hour in seven seconds. Units of Acceleration The SI units for acceleration are m / s2(meters per second squared or  meters per second per second). The gal or galileo (Gal) is a unit of acceleration used in gravimetry but is not an SI unit. It is defined as 1 centimeter per second squared. 1 cm/s2 English units for acceleration are feet per second per second,  ft/s2 The standard acceleration due to gravity, or standard gravity  g0 is the gravitational acceleration of an object in a vacuum near the surface of the earth. It combines the effects of gravity and centrifugal acceleration from the rotation of the Earth. Converting Acceleration Units Value m/s2 1 Gal, or cm/s2 0.01 1 ft/s2 0.304800 1 g0 9.80665 Newtons Second Law- Calculating Acceleration The classical mechanics equation for acceleration comes from Newtons Second Law: The sum of the forces (F) on an object of constant mass (m) is equal to mass m multiplied by the objects acceleration (a). F am Therefore, this can be rearranged to define acceleration as: a F/m The result of this equation is that if no forces are acting on an object (F   0), it will not accelerate. Its speed will remain constant. If mass  is added to the object, the acceleration will be lower. If  mass  is removed from the object, its acceleration will be higher. Newtons Second Law is one of the three laws of motion Isaac Newton published in 1687 in  Philosophià ¦ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy).   Acceleration and Relativity While Newtons laws of motion apply at speeds we encounter in daily life, once objects are traveling near the speed of light, the rules change. Thats when Einsteins special theory of relativity is more accurate. The special theory of relativity says it takes more force to result in acceleration as an object approaches the speed of light. Eventually, acceleration becomes vanishingly small and the object never quite achieves the speed of light. Under the theory of general relativity, the principle of equivalence says that gravity and acceleration have identical effects. You dont know whether or not you are accelerating unless you can observe without any forces on you, including gravity.

Sum of Squares Formula Shortcut

Sum of Squares Formula Shortcut The calculation of a sample variance or standard deviation is typically stated as a fraction. The numerator of this fraction involves a sum of squared deviations from the mean. In statistics, the formula for this total sum of squares is ÃŽ £ (xi - xÌ„)2 Here the symbol xÌ„ refers to the sample mean, and the symbol ÃŽ £ tells us to add up the squared differences (xi - xÌ„) for all i. While this formula works for calculations, there is an equivalent, shortcut formula that does not require us to first calculate the sample mean. This shortcut formula for the sum of squares is ÃŽ £(xi2)-(ÃŽ £ xi)2/n Here the variable n refers to the number of data points in our sample. Standard Formula Example To see how this shortcut formula works, we will consider an example that is calculated using both formulas. Suppose our sample is 2, 4, 6, 8. The sample mean is (2 4 6 8)/4 20/4 5. Now we calculate the difference of each data point with the mean 5. 2 – 5 -34 – 5 -16 – 5 18 – 5 3 We now square each of these numbers and add them together. (-3)2 (-1)2 12 32 9 1 1 9 20. Shortcut Formula Example Now we will use the same set of data: 2, 4, 6, 8, with the shortcut formula to determine the sum of squares. We first square each data point and add them together: 22 42 62 82 4 16 36 64 120. The next step is to add together all of the data and square this sum: (2 4 6 8)2 400. We divide this by the number of data points to obtain 400/4 100. We now subtract this number from 120. This gives us that the sum of the squared deviations is 20. This was exactly the number that we have already found from the other formula. How Does This Work? Many people will just accept the formula at face value and do not have any idea why this formula works. By using a little bit of algebra, we can see why this shortcut formula is equivalent to the standard, traditional way of calculating the sum of squared deviations. Although there may be hundreds, if not thousands of values in a real-world data set, we will assume that there are only three data values: x1 , x2, x3. What we see here could be expanded to a data set that has thousands of points. We begin by noting that( x1 x2 x3) 3 xÌ„. The expression ÃŽ £(xi - xÌ„)2 (x1 - xÌ„)2 (x2 - xÌ„)2 (x3 - xÌ„)2. We now use the fact from basic algebra that (a b)2 a2 2ab b2. This means that (x1 - xÌ„)2 x12 -2x1 xÌ„ xÌ„2. We do this for the other two terms of our summation, and we have: x12 -2x1 xÌ„ xÌ„2 x22 -2x2 xÌ„ xÌ„2 x32 -2x3 xÌ„ xÌ„2. We rearrange this and have: x12 x22 x32 3xÌ„2 - 2xÌ„(x1 x2 x3) . By rewriting (x1 x2 x3) 3xÌ„ the above becomes: x12 x22 x32 - 3xÌ„2. Now since 3xÌ„2 (x1 x2 x3)2/3, our formula becomes: x12 x22 x32 - (x1 x2 x3)2/3 And this is a special case of the general formula that was mentioned above: ÃŽ £(xi2)-(ÃŽ £ xi)2/n Is It Really a Shortcut? It may not seem like this formula is truly a shortcut. After all, in the example above it seems that there are just as many calculations. Part of this has to do with the fact that we only looked at a sample size that was small. As we increase the size of our sample, we see that the shortcut formula reduces the number of calculations by about half. We do not need to subtract the mean from each data point and then square the result. This cuts down considerably on the total number of operations.