Describing Motion: Kinematics in One Dimension

The purpose of this tutorial is to teach you problem solving strategies and skills. This will be the primary focus of the tutorials in all the early chapters. In particular, this is designed to help you work the Practice Problems on the Companion Website, as well as, the textbook problems.

Solving One-Dimensional Motion Problems

1. v = v_o + at
2. x = x_o + v_ot + 0.5at²
3. v² = v_o² + 2a(x-x_o)

t is the time at the end of the problem (assuming that t=0 at the start),

x_o is the location at the start of the problem (i.e. when t=0),

x is the location at the end of the problem,

v_o is the speed at the start,

v is the speed at the end of the problem,

a is the acceleration, assumed constant.

1. If the problem involves only one object undergoing one period of acceleration (as in practice problems 1 and 2)
   then construct the following table:

   xo	_
   x	_
   vo	_
   v	_
   a	_
   t	_

   
   
2. Fill in the table. You must find numbers (values) for four of the six variables to work the problem. The values for some of the variables may be obscured in the wording of the problem so don't get discouraged if the information seems hidden. The approach to take is the same you would use in reading a mystery story. You know that there are clues to the mystery's solution in the story itself , some are out in the open and some hidden. As a reader it is your job to find all the clues. Physics problems are much like mystery stories, you need information to solve them. Some of the information is obvious and some hidden. Use the following hints to help uncover the hidden information :
   
   • Unless stated otherwise, assume x_o = 0.
   • Words like rest, and dropped usually imply v_o = 0.
   • Words like stop, and stopped usually imply v = 0.
   • For an object tossed vertically upward, v = 0 at its highest point .
   
   

   Interactive Example

   Read the problem below then fill in the table of variables, you will be coached if you do something wrong.
   
   A car accelerates from rest to 100.8 km/h in 8 seconds. What is its acceleration?
   
   Enter the data below. After entering a number, click outside of the box each time. If you make a mistake you will be coached. If nothing happens, you got it right! Congratulations.

   xo
   x
   vo
   v
   a
   t

   When you have found four of the six needed values, use the equations as follows :

   1. If the variable you are looking for is on the left side of an equation, check to see if you know everything on the right side of the equation. If so, use this equation if not, try step 2.
   2. Identify which equations have the sought after variable, and check to see if all the other variables in either equation are known. If so, use this equation (and algebra) to obtain your answer if not, try step 3.
   3. Find an equation in which only one variable is unknown. Solve for this variable and enter it in the above table of variables. Then repeat steps 1, 2 and 3 until you obtain a value for the desired variable.

   Interactive Example

   Listed below are the three equations available to you. Choose the one you think you should use to find the acceleration!

   1. v = v_o + at
   2. x = x_o + v_ot + 0.5at²
   3. v² = v_o² + 2a(x-x_o)

   Answer:

   If there are two objects as in practice problem 4), or if there is one object undergoing two accelerations as in practice problem 8) you will need two tables. That is, you will need one table for each of the two objects or one table for each of the two accelerations of the single object. The two tables should look like the following:
   

   xo _ x _ vo _ v _ a _ t _

   x'o _ x' _ v'o _ v' _ a' _ t' _

   
   

   After you fill in the two tables check to see whether or not you know four of the values in one of the tables. If so, then use the equations to find the two unknowns in that table. Chances are good, then, that one of these calculated values will be a missing value in the other table (e.g., the final velocity in one table might be the initial velocity in the other). Read the problem carefully to discover these kinds of relationships.
   
   

   If neither table has four known elements then one of the elements in one table must equal the analogous element in the other table. For example, it may be true that v = v' or that x = x' at the end of the problem. This reduces the number of unknowns by one and gives you another equation to work with e.g., if v=v' then v_o + at = v'_o + a't' , or if x = x' then x_o + v_ot + at²/2 = x'_o + v'_ot' + a't'²/2 . This should allow you to finish the problem. If it does not, then there must be more information in the problem that will let you establish other relationships (equations). Read the problem carefully to look for information that you have not used.
   
   

   Note also that if the problem involves two objects moving at the same time as in practice problems 4 and 9, t will always equal t' in the above equations. If one object undergoes two sequential accelerations on the other hand (as in practice problem 8), then t will not equal t'. In this case, however, the final values for the velocity (v) of the first acceleration will usually be the initial value (v'_o) of the second acceleration period, v'_o = v.
   

   Interactive Example

   A car accelerates from rest to 100.8 km/h in 8 seconds, and then decelerates to rest in half the distance. What is its deceleration?
   
   This is an example of a problem with two accelerations, the problem statement gives you values for six of the 12 variables. Enter them below (don't forget to click outside each box each time, as before).

   xo x vo v a t

   x'o x' v'o v' a' t'

   
   If you have only found five of the six elements you can get a hint  HERE! You are free to take x'o as zero or if you want you can set it equal to x. In which case, you have only five known elements but one more equation namely x'o = x'. .)

   
   
   Once you know six of the 12 variables and you can use two of the equations from each block below to find values for four additional variables, for a total of 10. You need the value for two more variable to completely solve the problem. However, two more variable values are not available.

   v = vo + at x = xo + vot + 0.5at2 v2 = vo2 + 2a(x-xo)

   v' = v'o + a't' x' = x'o + v'ot + 0.5a't'2 v'2 = v'o2 + 2a'(x'-x'o)

   
   
   
   
   

   1. The first part of the problem is the same as the first interactive example above, so the remaining unknown variables, a and t, in part 1 can be determined using the procedure outlined in step II.
   2. When that is done you have determined the values for eight of the variables. There are two equations remaining to use from the three on the right above. This raises the total number of potentially known variables to 10.
   3. You need two more equations.
   4. To find one of them review VI above.
   5. Finally, read the problem statement carefully to see if there is any information present you have not used .
   6. The changes are good that the unused information will furnish the needed equation.
   7. In order to calculate the deceleration (a'), you need to know the stopping distance, which is not directly known. You know the distance travelled in the first part of the problem, however, so the stopping distance can be directly calculated from the given information based on the problem statement, namely, d' = d/2.

   Interactive Example

   When you have determined the stopping distance (x'), use equation-picking strategy outlined earlier to choose one of the following three equations to solve for the deceleration (a'):

   1. v' = v'_o + a't'
   2. x' = x'_o + v'_ot' + 0.5a't'²
   3. v'² = v'_o² + 2a'(x'-x'_o)

   Almost all problems can be worked following the strategy outlined above. The secret to success is to be meticulous in following the steps as outlined.

   Remember:

   • x, v, a are vectors, so they have magnitude and a direction.
   • It is best to take right (or up) as positive and left (or down) as negative, so that the acceleration of gravity is always down (negative).
   • Convert all speeds to meters per second (m/s), all distances to meters (m), and all accelerations to m/s² when working the practice problems. In other cases, at the very least, convert all values to consistent units so that you don't introduce errors when you plug in numbers to calculate values.