Physics Classroom 1d Kinematics Problems

[Josette Werst]

Earlierin Lesson 6, four kinematic equations were introduced and discussed. A useful problem-solving strategy was presented for use with these equations and two examples were given that illustrated the use of the strategy. Then, the application of the kinematic equations and the problem-solving strategy to free-fall motion was discussed and illustrated. In this part of Lesson 6, several sample problems will be presented. These problems allow any student of physics to test their understanding of the use of the four kinematic equations to solve problems involving the one-dimensional motion of objects. You are encouraged to read each problem and practice the use of the strategy in the solution of the problem. Then click the button to check the answer or use the link to view the solution.

At this point, it appears that these problems seem to be quite long and take several steps. While that is an inherent part of physics in many ways, it will start to seem simpler as time goes on. This problem presents the perfect example. While it may have been easy to combine lines 4 and 5 mathematically, they were shown separately here to make sure the process was as clear as possible. While you should always show all of the major steps of your problem-solving process, you may find that you are able to combine some of the smaller steps after some time of working with these kinematic equations.

If we remember back at the beginning, we said that our final velocity would have to be less than our initial velocity because the problem statement told us that we were decelerating. Our initial velocity was 25\text m/s which is, indeed, greater than 5\text m/s so our answer checks out.

A ghost is sliding a wrench across a table to terrify the mortal onlooker. The wrench starts with a velocity of 2\text m/s and accelerates to a velocity of 5\text m/s over a distance of 7\text m. What acceleration did the ghost move the wrench with?

Kinematics is the branch of physics that studies the motion of objects without taking into account the causes of motion. It deals with concepts such as position, displacement, velocity, and acceleration.

Kinematics is important because it provides a framework for understanding and analyzing the motion of objects, which is crucial in many fields of science and engineering. It also helps us make predictions about future motion and can be used to design and improve technologies.

Some interesting kinematics lessons for physics teachers could include projectile motion, circular motion, and motion in one and two dimensions. Other topics such as relative motion, motion graphs, and free fall could also be engaging for students.

There are many ways to make kinematics lessons more interactive and engaging for students. Some ideas include using hands-on activities, demonstrations, simulations, and real-world examples. You could also incorporate group work, discussions, and problem-solving activities to encourage active learning and critical thinking.

Yes, there are many online resources available for teaching kinematics. Some popular websites include Khan Academy, Physics Classroom, and Physics Central. Additionally, there are various educational videos, simulations, and interactive activities that can be found on these websites and other online platforms.

There are 23 ready-to-use problem sets on the topic of 1-Dimensional Kinematics. The problems target your ability to use the average velocity and average acceleration equations, to interpret position-time and velocity-time graphs, and to use the kinematic equations to determine the answer to problems, including those which involve a free fall acceleration. Problems range in difficulty from the very easy and straight-forward to the very difficult and complex.

Position-time graphs represent variations in an object's position over the course of time. Thus, an inspection of the values being plotted along the vertical axis allows one to determine position values and changes in position values. These axis values are related to the distance and displacement for an object's motion. The displacement of an object is simply the overall change in position; it considers only the initial and the final positions and ignores all in-between positions. Being a vector quantity, it includes a direction that is often expressed as a positive or a negative sign (e.g., + for rightward and - for leftward). The distance traveled by an object is the accumulation of all the ground covered. In effect, it is the sum of all the changes in position for each leg of a trip; such sums must ignore the + and - signs since distance is a scalar quantity. Being a scalar quantity, it is ignorant of direction and does not distinguish between a change in position to the left or to the right. If a person walks 2 meters, left and then 6 meters right, then the displacement is 4 meters to the right and the distance is 8 meters.

Slopes of such position-time graphs yield a ratio of the position change to time change for any specified interval of time. As such, the slopes are equivalent to the velocity of an object. Velocity (slope in this case) is a vector quantity that has a direction associated with it. Mathematically, the velocity direction is often represented by a positive or a negative value. Upward-sloping lines are associated with positive velocities and downward-sloping lines are associated with negative velocities. For a complex motion with multiple slopes, the average velocity can be determined by connecting the initial and final position and calculating the slope of the connecting line. Alternatively, the average velocity could be determined by reading the overall position change (displacement) off the axis and dividing by the change in time.

The average speed for any motion is simply the distance to time ratio. For a simple motion involving a constant velocity, the average speed is the absolute value of the slope. For a complex motion with several slopes, the average speed is the overall distance traveled divided by the time of travel. As mentioned above, the overall distance traveled can be determined by summing the absolute value of the individual position changes for each leg of the motion.

Velocity-time plots represent variations in an object's velocity over the course of time. As such, the slope of such graphs provides a velocity change to time change ratio; this ratio represents the acceleration of the object. Some of the questions in these problem sets will test your ability to use the coordinates for two points on a line to determine the slope. The slope is simply the change in the y coordinates divided by the change in the x-coordinate. You can use our video to learn more about the process.

Velocity-time plots can also be used to determine the distance traveled by an object and the displacement of an object. The area between a plotted line and the time axis (which could be a positive or a negative value) represents the displacement of the object. The area spoken of here is formed by the plotted line, the time axis and two imaginary vertical lines drawn at two specified points in time. The resulting area could be represented by either a rectangle, a triangle, a trapezoid or a collection of such shapes. The graphic below depicts a variety of resulting shapes and the appropriate formulas for calculating areas from each of them.

For a complex motion like that represented below, the total area can be broken into a collection of smaller shapes. The area of the individual sections can then be added together to determine the total area.

Kinematics (the topic of the current unit) is the science of describing the motion of objects. An object's motion can be described using words, diagrams, numbers, graphs and equations. The most commonly used of all equations are the four kinematic equations - affectionately known as the big four. These four equations allow a student to make a prediction of how fast (velocity and speed), how far (displacement and distance), or how much time is required of an object during a motion. The four equations are listed below.

Each of the above kinematic equations have four variables. The usefulness of the equations is that they allow a person to make a prediction about the value of one of the variables if given the value of three other variables. By knowing three, one can calculate a fourth. The problem-solving strategy used in this collection of problems will center around this idea. Each problem consists of a word-story problem in which information about an object's motion is given. The goal is to carefully read through each story problem to identify at least three pieces of known information in order to calculate a fourth requested piece of information. Often the known information is explicitly stated - "A car moving with an initial velocity of 23.4 m/s...". At other times there are statements included within the word problem such as "Starting from rest, ...? or "...comes to a stop." Such statements imply that the initial velocity is 0 m/s and the final velocity is 0 m/s (respectively).

While the equations are extremely useful, there is one condition which must be met in order for the equations to be used. The object under study must have a constant and uniform acceleration. If an object changes its acceleration at a given point during the motion such that it accelerates at one rate and then later accelerates at a second rate, then the motion must be divided into two phases and each phase must be analyzed separately.

Some of the problems in these problem sets involve free fall type motion. Free fall is a type of motion in which the only force acting upon the moving object is the force of gravity. When an object is in free fall, its motion is influenced solely by the force of gravity and the object accelerates at 9.8 m/s/s. The 9.8 m/s/s value is the acceleration of any free falling object irregardless of the characteristics of the object. Such an acceleration is dependent solely upon the gravitational characteristics of the planet and is thus called the acceleration of gravity or acceleration caused by gravity and represented by the variable g. While the value of g is 9.8 m/s/s on Earth, it is a different value on other planets where the gravitational characteristics are different. NOTE: Some classes use a rounded value of 10 m/s/s to simplify calculations.)

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