

Lesson 1: Describing Motion with WordsIntroduction to the Language of Kinematics 
Lesson 6: Kinematic Equations and ProblemSolvingKinematic Equations and ProblemSolvingThe four kinematic equations which describe the mathematical relationship between the parameters which describe an object's motion were introduced in the previous part of Lesson 6. The four kinematic equations are:
In the above equations, the symbol d stands for the displacement of the object. The symbol t stands for the time for which the object moved. The symbol a stands for the acceleration of the object. And the symbol v stands for the instantaneous velocity of the object; a subscript of i after the v (as in v_{i}) indicates that the velocity value is the initial velocity value and a subscript of f (as in v_{f}) indicates that the velocity value is the final velocity value. In this part of Lesson 6 we will investigate the process of using the equations to determine unknown information about an object's motion. The process involves the use of a problemsolving strategy which will be used throughout the course. The strategy involves the following steps:
The use of this problemsolving strategy in the solution of the following problem is modeled in Examples A and B below.
The solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step involves the identification and listing of known information in variable form. Note that the v_{f} value can be inferred to be 0 m/s since Ima's car comes to a stop. The initial velocity (v_{i}) of the car is +30.0 m/s since this is the velocity at the beginning of the motion (the skidding motion). And the acceleration (a) of the car is given as  8.00 m/s^{2}. (Always pay careful attention to the + and  signs for the given quantities.) The next step of the strategy involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the car. So d is the unknown quantity. The results of the first three steps are shown in the table below.
The next step of the strategy involves identifying a kinematic equation which would allow you to determine the unknown quantity. There are four kinematic equations to choose from. In general, you will always choose the equation which contains the three known and the one unknown variable. In this specific case, the three known variables and the one unknown variable are v_{f}, v_{i}, a, and d. Thus, you will look for an equation which has these four variables listed in it. An inspection of the four equations above reveals that the equation on the top right contains all four variables. Once the equation is identified and written down, the next step of the strategy involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. 0 m^{2}/s^{2} = 900 m^{2}/s^{2} + (16.0 m/s^{2})*d (16.0 m/s^{2})*d = 900 m^{2}/s^{2}  0 m^{2}/s^{2} (16.0 m/s^{2})*d = 900 m^{2}/s^{2} d = (900 m^{2}/s^{2})/ (16.0 m/s^{2}) d = (900 m^{2}/s^{2})/ (16.0 m/s^{2}) d = 56.3 m The solution above reveals that the car will skid a distance of 56.3 meters. (Note that this value is rounded to the third digit.) The last step of the problemsolving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. It takes a car a considerable distance to skid from 30.0 m/s (approximately 65 mi/hr) to a stop. The calculated distance is approximately onehalf a football field, making this a very reasonable skidding distance. Checking for accuracy involves substituting the calculated value back into the equation for displacement and insuring that the left side of the equation is equal to the right side of the equation. Indeed it is!
Once more, the solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step of the strategy involves the identification and listing of known information in variable form. Note that the v_{i} value can be inferred to be 0 m/s since Ben's car is initially at rest. The acceleration (a) of the car is 6.00 m/s^{2}. And the time (t) is given as 4.10 s. The next step of the strategy involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the car. So d is the unknown information. The results of the first three steps are shown in the table below.
The next step of the strategy involves identifying a kinematic equation which would allow you to determine the unknown quantity. There are four kinematic equations to choose from. Again, you will always search for an equation which contains the three known variables and the one unknown variable. In this specific case, the three known variables and the one unknown variable are t, v_{i}, a, and d. An inspection of the four equations above reveals that the equation on the top left contains all four variables. Once the equation is identified and written down, the next step of the strategy involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. d = (0 m) + 0.5*(6.00 m/s^{2})*(16.81 s^{2}) d = 0 m + 50.43 m d = 50.4 m The solution above reveals that the car will travel a distance of 50.4 meters. (Note that this value is rounded to the third digit.) The last step of the problemsolving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. A car with an acceleration of 6.00 m/s/s will reach a speed of approximately 24 m/s (approximately 50 mi/hr) in 4.10 s. The distance over which such a car would be displaced during this time period would be approximately onehalf a football field, making this a very reasonable distance. Checking for accuracy involves substituting the calculated value back into the equation for displacement and insuring that the left side of the equation is equal to the right side of the equation. Indeed it is!
The two example problems above illustrate how the kinematic equations can be combined with a simple problemsolving strategy to predict unknown motion parameters for a moving object. Provided that three motion parameters are known, any of the remaining values can be determined. In the next part of Lesson 6, we will see how this strategy can be applied to free fall situations. Or if interested, you can try some practice problems and check your answer against the given solutions. 

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