Acceleration

Average acceleration is defined as the change in speed during a time interval divided by the length of the time interval or

Keep in mind that you do not use speeds and times in this equation.  You must use changes in speed and time intervals.  Be sure to use speed changes and time intervals that correspond to each other and also to the problem.  For example, if in a certain problem you were asked to find the average acceleration between 6s and 8s, do not divide the speed at 8s by the time since the start.  You must subtract the speed at 6 sec from the speed at 8 sec and divide the result by Δt which is 2s.  Whenever calculating ΔX, Δt, or ΔV always subtract the initial value from the final value and pay attention to the sign obtained, it has physical significance.

Uniform Acceleration

There are many situations in physics where an object   moves with a constant or nearly constant acceleration.  The following equations can be used in these situations

Be sure to use the previous  equations  only if the acceleration is constant.  Under some circumstances certain simplifications may be made. For example if the time = zero when the motion starts then you may replace Δt by t.  Similarly if the motion  starts at the origin of the coordinate system  ΔX may be replaced by X.  If the object is at rest when the motion starts all terms involving V_i may be dropped.  Pay careful attention to all algebraic signs as they mean something physically.  If a speed is positive that means that the object is moving in the direction you called positive, usually to the right on a horizontal reference frame or up in a vertical reference frame.  If an acceleration is positive the object is accelerating in the positive direction, if it is negative the object is accelerating in the negative direction.  Do not assume that a negative acceleration means that the object is slowing down.  If an object has a positive speed and a negative acceleration it is indeed slowing down, but if the speed is negative and the acceleration is also negative the object is speeding up in the negative direction.  In general, if the signs of the speed and the acceleration are the same then the object is speeding up in the direction indicated by the common sign.  If the signs are opposite then the object is slowing down.  If an object has an acceleration which is opposite its speed and maintains that acceleration, it will come to rest and then begin to move in the opposite direction at a constantly increasing speed.

Motion in Free Fall

Any object falling near the surface of the earth in a vacuum has a constant acceleration of 9.8 or 32.2 .  This motion is called free fall and the constant acceleration is called the acceleration of gravity and is denoted by either a "g" or an "a_g".  If a relatively compact object falls near the surface of the earth and its speed is low it approximates free fall quite well.  As an object gains speed, air friction becomes important and the acceleration no longer is equal to a_g.  If the motion of the object approximates free fall the equations given above may be used in solving problems with the appropriate value of a_g used for the acceleration.  
    

Solving Problems with Uniform Acceleration

There are many kinds of uniform acceleration problems and the only way to become proficient in solving them is to practice.  A few general suggestions will be given and sample problems solved but there is no substitute for practice, therefore  a large number of problems will be given for the student to solve.   The following steps might be followed in solving problems involving uniform acceleration:

1. Read the problem carefully, two or three times if necessary, to be sure you know what is to be found.
2. Make a list of all quantities given.  Use the same symbols that are used in the equations.  Be sure to include the units.  If the units are not consistent make any necessary changes now.  For example if the distance is given in meters and the speed in cm/sec then either change the distance to cm or the speed to m/sec.
3. Look for any quantities whose values are implied by the problem but not  stated explicitly.  For example: If a body is in free  fall its acceleration will be the acceleration of gravity; if it is dropped its initial speed will be  0.   The following hold for an object which is thrown vertically upward: Its speed at the very  top of its path is 0, the time required for it to reach its highest point is  equal to the time to return to the same level, and the upward speed at the  start is equal in magnitude to the downward speed when it returns to the same level.
4. Decide on a reference frame.  Sketch a diagram if it will help in any way.  Set it up to make the problem as simple as possible.  For example if an object  is dropped from a high building you might decide to call the point at which it  was started  0 and call the downward direction positive.  In this case both the speed and acceleration will be positive.  However, if an object is thrown upward from  the ground it is probably simpler to set the origin on the ground and call upward positive.   Remember, in this case the acceleration of gravity which is always downward will be negative for the entire motion of the object.
5. Be sure you know what you want to find.  Write it down if necessary again using the same symbol(s) used in the equations.
6. Now examine the equations to see if one contains the desired unknown and you know the values for all other terms in the equation.  If so, use it to solve the problem.  You may have to rearrange the equation to solve for the desired unknown, it is too much to expect that the term you want will be nicely located by itself on one side of the equation.
7. You may not be able to find the desired quantity immediately.  If you can't, use the equations to find any unknown quantities, one will probably help you to find the one you want.  If the problem has several parts you will frequently use the answer for one part to determine the answer for a later part.  However, don't assume that problems must always be done in the order stated.  Sometimes it is easier to change the order of the problem.
8. When you are done be sure to indicate the answer clearly on your paper. Always show the correct units,  failure to do so will cause loss of credit.   Don't become discouraged if the problems seem hard at first, as you practice they will become easier.  There will come a time when you go back and look at some of the first problems you solved and wonder why you had such a hard time with them.

Editor's Note: The problems in this section are designed to be readily solvable simply using one of the equations above. More complex problems will appear in the next section.

1.

If a car accelerates from 20 to 40 in 4 seconds, what is the average acceleration?

2.

In the previous question, assuming a constant acceleration, what is the average velocity?

3.

A car accelerates at 3 for 5 seconds. If the initial velocity was 10 , what is the final velocity?

4.

A car has an initial velocity of 20 , and accelerates at 2 while traveling 100 m. What is the car's final velocity?

5.

A car has an initial velocity of 0, an acceleration of 5 , and it accelerates at this rate for 10 seconds. What distance is traveled?

6.

When a car starts at rest, what quantity does that information imply?

7.

If a ball is thrown upward, what can you conclude about its velocity when it reaches its highest point?

8.

Aside from losing credit, why do you think it is important to show units in an answer?

9.

An automobile traveling at a speed of 3 0mi/hr accelerates uniformly to a speed of 60 mi/hr in 10 sec. How far does the automobile travel during the acceleration?

10.

During takeoff a Boeing 747 jumbo jet is accelerating at 8.0m/s². If it requires 25 sec to reach takeoff speed, determine the takeoff speed and how far the jet travels on the ground.

11.

Suppose we designate up as the positive direction and that we use a coordinate system in which the origin is at the ground level. State the sign of the velocity and the acceleration of: an elevator rising and coming to rest; an elevator at the instant it starts to descend; an elevator descending and coming to rest; an elevator rising at constant speed