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Guide 2-1a. Solving Kinematics Problems with the DVAT Equations

Table 2-4 (p. 34) in your text gives a set of equations that apply to objects undergoing uniform acceleration. These can be used to solve one of the three major problem types that we'll encounter in mechanics. We'll call this type a dvat problem, where the acronym stands for position-velocity-acceleration-time. The equations are listed below with descriptions.

 No. Equation Description 1 v = vo + at This is simply an algebraic rearrangement of the definition of acceleration:  a = Δv/Δt. Galileo defined acceleration in this way, because he found that objects rolling down inclines increased in speed by equal amounts in equal time intervals. By defining acceleration as a = Δv/Δt, he could say that these objects had constant acceleration. While he could have selected some other formula such as a = Δv/Δx, that wouldn't have been useful in describing the motion of real objects. 2 vav = ½(vo + v) This is called the Merton Theorem and was developed at Merton College of Oxford in the 13th century. Basically, it amounts to saying that when a quantity increases at a uniform rate, the average value of the quantity is equal to the middle value. For example, suppose you had the series of numbers 1,2,3,4,5. Add them up and divide by 5 to get 15/5 = 3, the average. But this is also the middle number in the sequence. Note that the numbers in the sequence increase by equal amounts. Try applying the Merton Theorem to the sequence 1,3,4,5, and you'll see it doesn't work for that non-uniform sequence. 3 x = xo + ½(vo + v)t This is a combination of the second equation and the definition of average velocity, vav = Δx/Δt. 4 x = xo + vot + ½at² These equations are derived algebraically from combinations of the first three equations. Note that the sixth equation is one that isn't in the book. However, it can be useful in solving some problems. 5 v² = vo² + 2a(x - xo) 6 x = xo + vt - ½at²

Whenever using the dvat equations, keep in mind the following:

1. The quantities subscripted with o indicate initial values, while those that don't have subscripts represent final values. The initial value of the time is assumed to be zero.

2. Times are always positive or zero.

3. The equations apply only to objects with constant or zero acceleration.

4. The equations apply to objects moving in a straight line.

5. The quantities x, v, a, and their initial values may be positive or negative. The sign is determined by the direction selected for the positive x-axis. Suppose the positive x-axis is set up to be pointing to the right. In this case, an object moving to the right has a positive velocity, and an object moving to the left has a negative velocity. Consider the following four possible situations.
 For all of the following situations, the direction of positive displacement, x, is defined to be to the right:  +x +x Description of the motion in words Sign of v Direction of v Sign of a Direction of a Description of the motion in mathematical terms Object is moving to the right and the magnitude of the velocity is increasing (object is speeding up to the right) + ---> + ---> The velocity is becoming more positive, so Δv in Δv/Δt is positive. Object is moving to the right and the magnitude of the velocity is decreasing (object is slowing as it moves to the right) + ---> - <--- The velocity is becoming less positive, so Δv in Δv/Δt is negative. Object is moving to the left and the magnitude of the velocity is increasing (object is speeding up to the left) - <--- - <--- The velocity is becoming more negative, so Δv in Δv/Δt is negative. Object is moving to the left and the magnitude of the velocity is decreasing (object is slowing to the left) - <--- + ---> The velocity is becoming less negative, so Δv in Δv/Δt is positive.
1. As is frequently the case in physics, it's not necessarily useful to memorize the above statements. If you understand where the relationships come from, then you can produce them. That's why we provide the descriptions in mathematical terms above to help you understand how the sign of the acceleration is determined.

2. Something important to realize is that a negative acceleration needn't mean that an object is going slower and slower. The negative sign is determined by the choice of positive x, which in turn determines the direction of positive v.

With the above in mind, we'll look at a simple strategy to solve dvat problems. We'll describe the strategy first and then apply it to a specific situation.

 Strategy to solve dvat problems Step 1.  After reading the problem, draw a diagram. On the diagram, indicate the origin, the direction you select for +x, and the corresponding directions for initial velocity and acceleration. Label any other relevant quantities. Step 2.  List all the given information. Identify the givens with the same symbols that are used in the dvat equations, namely, x, xo, v, vo, vav, and t. Given the direction you selected for +x, make sure all the given information has the correct sign. Step 3.  State the unknown that you're supposed to find. Step 4.  Look at the list of dvat equations and select one for which all quantities are known except for the unknown that you're solving for. Step 5.  Algebraically solve the dvat equation you selected for the unknown. That means to solve in symbolic form without numbers. Step 6.  Substitute the given values with units. Do the arithmetic to arrive at the final answer. Step 7.  Apply sign, units, and sensibility checks.

Now let's apply the strategy to a problem. Click here to go on.