Choosing the Appropriate Kinematics Equation
In the study of motion in physics, there are five kinematics equations that can be used to solve problems involving constant acceleration. They are:
V = Vo + at
d = 1/2*(Vo + V)*t
d = Vo*t + 1/2*at^2
d = V*t – 1/2*at^2
V^2 = Vo^2 + 2ad
Notice that each equation contains exactly four of the five kinematics variables (Distance d, Initial velocity Vo, Final velocity V, Acceleration a, and Time t). The first equation is missing d, the second equation is missing a, the third equation is missing V, the fourth equation is missing Vo, and the fifth equation is missing t.
Students often have a hard time deciding which equation to choose for solving a problem, so this sample problem will show how to choose the appropriate equation and solve the problem.
An airplane, starting from rest, accelerates at 4 m/s^2 to a speed of 100 m/s, which is required for takeoff. What length of runway is necessary?
First, identify three variables that are known and write them down, and identify the variable you are asked to solve for, and write it down with a question mark. For this problem, we have:
Vo = 0 (since the airplane starts from rest)
a = 4 m/s^2
V = 100 m/s
d = ?
Since d is unknown, students often make the mistake of trying to use the kinematics equation that is missing d. However, the equation should be chosen based on the variable that is completely missing from the above list, which in this case is t. We don’t know what t is, and we’re not asked to find t. Looking at the list of kinematics equations, it is the fifth equation that is missing t, so we’ll use that one.
Start by writing down the equation:
V^2 = Vo^2 + 2ad
If any variable in the equation is zero, remove it. In this case, Vo = 0:
V^2 = 2ad
Rearrange the equation to get the unknown variable by itself. In this case, we are solving for d, so divide both sides by 2a:
V^2/2a = d
Finally, plug the numbers in:
d = 100^2 / (2*4) = 10,000 / 8 = 1250 m or 1.25 km
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