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2D Kinematics: Tips & Tricks

2D Kinematics: Tips & Tricks

Problem Solving Guide

Algorithm


1. Read and re-read the whole problem carefully.

2. Visualize the scenario.  Mentally try to understand what the object is doing.

   a. Motion diagrams are a great tool here for visual cues as to what the motion of an object looks like.

3. Draw a physical representation of the scenario;  include initial and final velocity vectors, acceleration vectors, position vectors, and displacement vectors.

4. Define a coordinate system;  place the origin on the physical representation where you want the zero location of the x and y components of position.

5. Identify and write down the knowns and unknowns.

6. Identify and write down any connecting pieces of information.

7. Determine which kinematic equation(s) will provide you with the proper ratio of equations to number of unknowns; you need at least the same number of unique equations as unknowns to be able to solve for an unknown.

8. Carry out the algebraic process of solving the equation(s).

    a. If simple, desired unknown can be directly solved for.

    b. May have to solve for intermediate unknown to solve for desired known.

    c. May have to solve multiple equations and multiple unknowns.

    d. May have to refer to the geometry to create another equation.

    e. If multiple objects or constant acceleration stages or dimensions, there is a set of kinematic equations - something will connect them.

9. Evaluate your answer makes sense, are the units and dimensions correct and the results are within reason.

Misconceptions & Mistakes

  • An object's acceleration does not determine the direction of motion of the object.
  • When an object is thrown upwards at an angle near the surface of the earth, the y-component of velocity is zero at the maximum height that the object reaches, the acceleration is not zero.
  • Remember that the velocity in one direction does not imply velocity in another.
  • The displacement of an object does not depend on the location of your coordinate system.
  • The final velocity of an object when dropped from some height above ground is not zero.  The kinematic equations do not know that the ground is there, thus the final velocity is the velocity of the object just before it actually hits the ground.

Pro Tips

  • Draw a physical representation of the scenario in the problem.  Include initial and final velocity vectors, displacement vector, and acceleration vector.
  • Place a coordinate system on the physical representation.  Location of the origin does not matter, but some locations make the math easier than others (i.e. setting the origin at the initial or final location sets the initial or final position to zero).
  • Always identify and write down the knowns and unknowns.
  • Be very strict with labeling the kinematic variables;  include subscripts indicating which object they are associated with, the coordinate, information about what stage if a multi-stage problem, and initial/final identifications.  Doing this early on will help avoid the common mistake of misinterpreting the meanings of the variables when in the stage of algebraically solving the kinematic equations.
  • Remember in 2D kinematics problems is that the two dimensions are entirely independent of each other. Treat them separately, come up with the knowns and unknowns for all variables in each direction.
  • Keep track of your angles and break all forces in various directions into their components in your chosen coordinate system. 

Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways. 

Physical

Physical Representations describes the physical phenomena of the situation in a visual way.

 

Kinematics problems are often analyzed with physical descriptions. Below we have an object launched from a ledge with initial horizontal and vertical velocity components. The vectors indicate the relative magnitudes of the horizontal and vertical velocity components at different stages of the objects motion.

This is picture of an object undergoing projectile motion. The figure shows the object at various times in it's flight. There are vectors to represent the velocity, acceleration, and change in position for the motion

In addition, simple 1D motion diagrams are utilized to describe the speed of an object and whether or not the object has any acceleration. 

This image shows three long strips of paper with black dots in different position. The black dots is a representation of a position over time. In the first strip of paper, the black dots are evenly spaced apart from each other. There is also a person with a speech bubble next to it that says constant speed. The second strip of paper shows the black dots with increasing distance between each other over time. There is a person with the speech bubble next to it that says accelerating. The third strip of paper shows fewer black dots with more increasing distance between each other over time. There is a person with the speech bubble next to it that says lots of acceleration.

Mathematical

Mathematical Representation uses equation(s) to describe and analyze the situation.

Main kinematics equations. Notice that we use $ r $ to represent the vector position. In the videos below, Professor Matt Anderson uses $ x $ and $ y $ to denote position in a given coordinate direction.

$ \vec{r}_{f} = \vec{r}_{i}+\vec{v}_{i} \Delta t + \frac{1}{2} \vec{a} \Delta t^{2} $

 $ \vec{v}_{f} = \vec{v}_{i} + \vec{a} \Delta t $

$ v_{fx}^{2} = v_{ix}^{2}+2a_{x} \Delta x $

A representation with the words kinematic equation for constant acceleration one (K.e.q. one) on the top. There is an equation that shows that the final position vector is equal to the initial position vector plus the initial velocity vector multiplied by time plus one-half the average acceleration vector multiplied by the change in time squared. This is also written in words below with a note that says this equation contains vector and scalar quantities.

A representation with the words kinematic equation for constant acceleration two (k.e.q. two). There is a equation that shows that the final velocity vector is equal to the initial velocity vector plus the average acceleration vector multiplied by the change in time. This is also written in words below with a note that says this equations contains vector and scalar quantities.

A representation with the words kinematic equation for constant acceleration three (k.e.q. three) on the top. There is an equation that shows that the square of the final velocity vector is equal to the square of the initial velocity vector plus two times the product of the average acceleration vector and the change in position. This is also written in words below with a note that says that this equation contains vector and scalar quantities.

Watch Professor Matt Anderson explain 1D and 2D Kinematics equations using the mathematical representation.

Video 1: Kinematics equations in 1D

Video 2: Kinematics equations in 2D

 

Graphical

Graphical Representation describes the situation through use of plots and graphs.

the Physics Classroom introduces the relationship between kinematics equations and their graphical representation. In addition, for a more in-depth discussion please refer to the graphical analysis section. 

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Descriptive

Descriptive Representation describes the physical phenomena with words and annotations.

  The branch of mechanics concerned with the motion of objects without reference to the forces that cause the motion.

Experimental

Experimental Representation examines a physical phenomena through observations and data measurement.

 

For example, a person throws a ball upward into the air with an initial velocity of $15.0 \frac{m}{s}$. You can calculate how high it goes and how long the ball is in the air before it comes back to your hand. Ignore air resistance.

A woman is throwing a ball vertically upwards with an initial point a in her hand, point b at the maximum height and a final point c back in her hand. on the left side as the ball is thrown up, there is a green arrow that points up that represents the velocity and a yellow arrow above it that represents gravity. at the top in point b, it is written that velocity equals zero. On the right side as the ball is falling back down, there is a green arrow that points down that represents the velocity and a yellow arrow above it that represents gravity.

 

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