Kinematics-Babykin/Steele

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ISCI 6501: Staci Babykin and Jason Steele

Distortion in a fluid. Image Credit: Professor M. Kostic, Northern Illinois University[1]

Grade Level and GPS

Grade Level Targeted: High School Physics

Science GPS Targeted:

SCSh1: Students will evaluate the importance of curiosity, honesty, openness, and skepticism in science.

a) Exhibit the above traits in their own scientific activities.

b) Recognize that different explanations often can be given for the same evidence.

c) Explain that further understanding of scientific problems relies on the design and execution of new experiments which may reinforce or weaken opposing explanations.

SP1: Students will analyze the relationship between force, mass, gravity, and the motion of objects.

a) Calculate the average velocity, instantaneous velocity, and acceleration in a given frame of reference.

c) Compare graphically and algebraically the relationships among position, velocity, acceleration, and time.

d) Measure and calculate the magnitude of gravitational forces.

Content

Background Information

The study of mechanical motion in physics is closely related to the analysis of four fundamental quantities: distance, time, velocity, and acceleration. These quantities are related to each other via four equations called the kinematic equations. These kinematic equations are valid for situations where the either the velocity or the acceleration is constant.

Average vs. Instantaneous Velocity. Animation Credit: The Physics Classroom [2]

Displacement

Distance is the scalar quantity which refers to how much "ground" an object has covered during its motion; it is the total amount covered.
Displacement is the vector quantity which refers to the difference in beginning and ending points.

Velocity

Velocity (v) is the rate of change in position or displacement(x).
Velocity is a vector quantity that has both a magnitude and direction.

Magnitude is the quantity (scalar) value of a vector (like velocity or displacement)
Direction is the direction in which the scalar magnitude of the vector is traveling

Average velocity (vavg) of an object can be calculated by dividing the change in displacement by the change in time (t): νavg = Δx/Δt.
The instantaneous velocity is the velocity of an object at a specific time interval.

One way to determine the instantaneous velocity is to construct a straight line that is tangent to the position vs. time graph at that particular instant.
Initial velocity (vi) and final velocity (vf) are specific instances of instantaneous velocity.

To algebraically calculate the initial and final velocities of an object that has a constant acceleration use the following kinematic equation: Δx=½(vi + vf)Δt.

Acceleration

Acceleration (a) is defined as the rate of change in velocity.

Acceleration is a vector quantity
Average acceleration can be calculated by dividing the change in velocity by the change in time: a = Δv/Δt

To algebraically calculate the displacement of an object with constant acceleration use the following equation: Δx=viΔt + ½a(Δt)^2
To algebraically calculate the final velocity of an object after any displacement use the following equation: vf^2=vi^2 + 2aΔx.

The equations above can be used to calculate the final velocity of almost any object that has a constant acceleration.

Free Fall

Neglecting air resistance, all free falling objects, on Earth, have the same constant downward acceleration of g; g is the acceleration due to gravity and has a value of of 9.81 m/s^2.

Since all falling objects have the same constant acceleration the above equations can be “edited” to find the vertical velocity (y-axis) of a falling object:

Δy=½(vi + vf)Δt

Δy=viΔt + ½g(Δt)^2

vf^2=vi^2 + 2gΔy

Projectile Motion

These kinematic equations can be used independently to solve for the one-dimensional motion of an object. However, if you have an object that is propelled into the air in a direction other than straight up or down, the velocity, acceleration, and displacement of the object do not all point in the same direction. This motion in two dimensions is referred to as projectile motion. By adding another dimension of kinematic equations, the solutions become more complex.

An example of projectile motion. Animation Credit: The Physics Classroom [3]

Instead of forcing the motion into a more complex two-dimensional form, the vectors from projectile motion are broken down into x- and y-components. Once the vectors are broken into the separate components, the simpler 1-D kinematic equations for both the x- and y-axes can be used to solve for the information from projectile motion. After solving each component separately, the values can be recombined to determine the resultant of the vector motion.

Integration of Science

The following lab relates to kinematics and uses the kinematic equations of displacement, velocity, and acceleration. The lab also applies the concept of projectile motion, because there is an object (a toy car) falling through the air after exiting a ramp with some velocity. Due to this two-dimensional motion, students must solve for the resultant of two-dimensional motion vector.

Teaching Considerations

Common Misconceptions

Average velocity is equal to the final velocity when the object is accelerating.

Students like to use the average velocity equation, vavg=Δx/Δt, to solve for an instantaneous velocity (such as final velocity) of an object that is accelerating.

Students need to understand that the average velocity equation can only be used to solve for an instantaneous velocity (final velocity) if the velocity of the object is constant--with no acceleration.

The horizontal (x-axis) component of the velocity of a projectile is not affected by the free fall acceleration of gravity, when the horizontal component is perpendicular to the ground and vice-versa.

Students have a hard time believing that if a bullet is shot out of a gun perpendicular to the ground at the same time a bullet is dropped straight down from the same height as the gun, that both bullets will hit the ground at the same time (neglecting air resistance).'

Inquiry Lab

Have the student build a ramp, where the end of the ramp is even with the end of the lab table.

The students’ objective is to find the velocity of which the object leaves the ramp, when the top of the ramp is at different heights.

After they have created a graph and have “tested” several different heights place a target on the floor and have the student adjust the height of the ramp so that the object will land on the target. They should be able to hit the target on the first try.

Teacher “Help Hints”:

Remind them that average velocity is not the same as final velocity when the object is accelerating.
Remind them that the horizontal (x-axis) component is not affect by gravity and vice-versa.
They will have to make a graph of height of ramp vs. final velocity of the object.
Have them first calculate how much time the object will stay in the air before in hits the ground.
Remind them once the object leaves the ramp that the object has a constant velocity in x-axis direction.
Make sure the students have access to meter sticks, timers, calculators and the kinematics equations.

Explanations of Misconceptions Addressed

Students cannot calculate the final velocity at which the object leaves the ramp by using the average velocity equation, because the object is accelerating.

The students have to understand that since the object leaves the table perpendicular to the ground that the horizontal (x-axis) component and the vertical component (y-axis) do not affect each other.

The students have to find the final velocity that the object leaves the ramp by first solving how much time the object stays in the air by using the following equation: Δy = viΔt + ½ g(Δt)^2.

The height of the table is equal to Δy, vi = 0, and g = 9.8 m/s^2.

Once the students know how much time the object is in the air, then they can calculate the velocity in the x-direction by using the average velocity equation because the velocity in the x-direction is constant.

They will use the vavg=Δx/Δt, the Δt is equal the time you calculated above and the Δx is the distance from the edge of the table to point at which the car hits the ground.

The procedure needs to be repeated at several different ramp heights and have the students create a graph of final velocity at which the object leaves the table vs. distance the objects travels when it hits the ground.

After graphing the data, the students will then draw a best fit line to help determine at what height the ramps needs to be placed in order to hit a target on the ground.

Resources

Teacher Internet Sources

An article discussing the benefits of teaching kinematics last--or "upside down"

Tevlin, R. (2007). "Upside Down" Physics: Teach Kinematics Last!. Retrieved September 12, 2009, from The Crucible Online. Website: "http://virtuallibrary.stao.ca/cruciblearticles"

"The Physics Classroom Tutorial" is a website that walks the user through various physics concepts. The "link" is to the kinematics contents page. Also available on this site are interactive shockwave and multimedia activities.

The Physics Classroom. (2009). 1-D Kinematics. Retrieved September 12, 2009, from The Physics Classroom Website: "http://www.physicsclassroom.com"

"The Physics Teacher Online" publishes papers on the teaching of physics, and on topics such as contemporary physics, applied physics, and the history of physics—all aimed at the introductory-level teacher.

The Physics Teacher Online. (2009). Retrieved September 13, 2009, from The Physics Teacher Online Web site: http://scitation.aip.org/tpt/

"American Association of Physics Teachers" is the professional organization for physics teachers and scientists who are dedicated to developing an understanding and appreciation for physics through the art and science of teaching.

The American Association of Physics Teachers. (2009). Retrieved September 13, 2009, from The American Association of Physics Teachers Website: http://www.aapt.org/aboutaapt/

A clip on YouTube from the movie "Monty Python and the Holy Grail". The clip shows a cryptic, old bridge keeper asking for the velocity of an unladen swallow. This would be a funny way to open a class introducing the concept of velocity.

Works Cited

Prof. M Kostic, Fluid Dynamics Course Website [4]