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Unit 6: 1-Dimensional Kinematics

Problem Set E

Overview:

Problem Set E targets your ability to use the kinematic equations to analyze physical situations involving motion in one dimensions. Some of the problems involve relatively simple free fall type motion (#1-9); others involve rather complex analysis which at times include two objects or several stages (#10-15). In order to solve these 15 problems, you will have to master the use of the four kinematic equations, understand the characteristics of an object in free fall and practice the habits of an effective problem solver.

The Big Four

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.

 d = vo • t + 0.5 • a • t2 vf = vo + a • t vf2 = vo2 + 2 • a • d d = (vo + vf) / 2 • t

 where d = displacement t = time a = acceleration vo = original or initial speed vf = final speed

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.

Free Fall

Some of the problems in this set 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 it 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 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 graviational characteristics are different.

There are some unique aspects about the trajectory of a free falling object which are worth noting and serve very useful in one's approach to solving problems. Any object which is originally projected vertically upward and under the sole influenc of gravity will eventually reach a peak height before turning around and falling back downward. At the peak of the trajectory, the velocity of the object is 0 m/s. The time required to reach the peak is equal to the time required to fall from the peak back to its original position. The total time of flight is thus twice the time required to reach the peak. The velocity of the object one second prior to the peak is of the same magnitude as the velocity of the object one second after reaching the peak. And similarly, the velocity of the object two seconds prior to the peak is of the same magnitude as the velocity of the object two seconds after reaching the peak.

To be successful on this problem set, you will need to be able to:

• implement a proper problem-solving strategy in which you identify known and unknown (requested) variables using the symbols of the kinematic equations.
• give consideration to units in such a manner that the units will properly cancel when the various performing algebraic operations dictated by the equations.
• diagram a physical situation (on the complex analyses) and use the kinematic equations to generate an equation which expresses the mathematical relationships described by the problem statement.

The following pages from The Physics Classroom tutorial may serve to be useful in assisting you in the understanding of the concepts and mathematics associated with these problems.

The Kinematic Equations | Problem-Solving | Sample Problems and Solutions

View Sample Problem Set.

 Problem Description Audio Link 1 Calculation of the peak height of a thrown ball from the knowledge of the original speed. 2 Referring to the previous problem; calculation of the time to reach the peak. 3 Calculation of the depth of a well from knowledge of the time a pebble takes to fall to the bottom. 4 Calculation of the original speed of a liquid from knowledge of the distance it rises to the peak of its trajectory. 5 Calculation of the original speed of a ball from knowledge of the height to which it rises to its peak. 6 Calculation of the time it takes a falcon to free fall a given distance to the location of a pigeon. 7 Calculation of the final speed of a falling camera from the original speed and the distance of fall. 8 Calculation of the final speed of a falling person if given the distance of fall. 9 Calculation of the original speed of a basketball player from knowledge of his vertical leap height. 10 Complex analysis of a biker's three-stage motion; must determine the average speed of the entire motion. 11 Calculation of the average speed of an arrow during the third second of motion after being launched with an original speed. 12 Complex analysis of a two-stage rocket; must calculate the height at a time after the second stage has burned out. 13 Calculation of the earliest time at which fireworks are a given distance above the ground if given their original velocity. 14 Complex analysis of the motion of two cars; must calculate the time at which one car (decelerating from a high speed) is side-by-side a slower car traveling at constant speed. 15 Complex analysis of the motion of two cars; must calculate the time at which one car (moving at a constant medium speed) is side-by-side another car which accelerates from rest to a high speed and then maintains the high speed.

Audio Help for Problem: 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15

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