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Sample Kinematics Problems with Solutions
Following are a variety of problems involving uniformly accelerated motion along a line. In the solution a list of known quantities will be given followed by a list of quantities wanted. The equations to be used will be identified by number from the list below, but the algebraic work of solving the equations will be left to the student. In most of the problems there will be a discussion of alternate methods of solution or suggestions and hints as to how to attack this particular type of problem. Do not try to just read this section. You must use pencil, paper and a calculator to obtain the maximum value from studying these problems.
Sample Problem #1
A car starts from rest and accelerates with a uniform acceleration of 10 ft s 2 . Calculate how long it will take for the car to reach a speed of 90 ft s (slightly over 60 mph), and calculate the distance the car moves during this time.
Sample Solution #1
Given V i 0 a 10 ft s 2 v f 90 ft s
Needed ΔX Δt
This problem can be solved by a straight forward application of equation 4 to find Δt and equation 5 to find ΔX. The student should demonstrate that the correct answers are Δt=9 s and ΔX=405 ft. It is a good idea to try to find the answers using slightly different procedures whenever possible. For example, in this problem a quick check can be made by using your calculated value for Δt, from equation 4 to find the average speed and then set X=V avg Δt to get 405 ft a different way. A second different method would be to use equation 6 and the value of Δt obtained from equation 4 to calculate ΔX. However, while both of the alternate techniques are good checks on the validity of your calculations it is best to avoid using an answer to one part as a basis for a second calculation whenever possible.
Sample Problem #2
A car moving at 30 m s stops with a constant acceleration in a distance of 100 m. Calculate the acceleration and the time to stop.
Sample Solution #2
Given V i 30 m s V f 0 m s Δx 100 m
Needed Δt a
To solve the problem apply equation 5 to find the acceleration (a=-4.5 m s 2 ). Use the calculated value of the acceleration to find the time (Δt=6.67 s) using either equation 4 or 6 (4 is easier).
Sample Problem #3
Explain the significance of the negative sign in problem 2.
Sample Solution #3
The acceleration is in the negative direction. Since V was arbitrarily assigned a positive direction, the acceleration must be in the opposite direction.
Sample Problem #4
Assume that the car in problem 2 keeps the same acceleration for an additional 5 seconds. Find its speed and position at the end of this time.
Sample Solution #4
Given V i 0 a -4.5 m s 2 Δt 5s
Needed V f X
Notice that here we are not asked for ΔX but are asked for X, the position at the end of the 5th second. We need to be sure to specify the position unambiguously, either with reference to the car's position at the start of problem 2 or with respect to its position at the end of problem 2 and the start of this problem. We will place the origin of the reference system at the position of the car at the start of problem 2. Thus at the end of problem 2 it has a position of X=100 m. Using equation 6 with the data listed above we see that ΔX= -56.25 m. The final position of the car is therefore 100 - 56.25 or 43.75 m. To find the speed use equation 4 and obtain a speed of - 22.5 m s . The complete answer to the problem could be stated as follows: "The car is 43.75 m from its starting point in the direction it was originally moving. It is moving back toward the starting point at a speed of 22.5 m s and is accelerating toward the starting point with an acceleration of 4.5 m s 2 ." This statement is much more meaningful than simply writing X = 43.75 m and V = -22.5 m s
Sample Problem #5
A baseball is batted vertically with an initial speed of 45 m s . Calculate: (a) the time before it returns to the level at which it started. (b) the maximum height the ball reaches. (c) the position, speed and acceleration 6 seconds after it is batted. (d) the position and speed 12 seconds after it is batted.
Sample Solution #5
Read this problem carefully and decide the order that you will do the problem. Keep in mind the facts mentioned in the hints such as: the time to reach the top is equal to one-half the total time in the air; the speed at the top is zero; and the speed when it reaches the same level is the same as the initial speed. In multi part problems such as this one, it will be necessary to list the given quantities for each part. As you solve the problem, check to be sure the answers to the various sections are consistent with each other. Before starting the problem decide which direction to call positive. In this problem we will call the upward direction positive and the downward direction negative.
Sample Problem #6
A ball dropped from the roof of a tall building passed a window ledge with a speed of 96 ft s and struck the ground 1.0 s later. (a) What is the height of the window? (b) How tall is the building?
Sample Solution #6
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