PYL 105

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Projectile motion

In this lab, we will study projectile motion, which is a special case of two-dimensional motion. In a two-dimensional space, an object's position is given by a pair of numbers (coordinates). There are two standard ways to represent such a position:

In Cartesian coordinates there are two orthogonal (at right angles) axes, usually called x and y. Imagine starting at the origin, you can reach a destination position by first moving along the x axis and then along the y axis. The destination point is identified as (x,y), where x is the distance moved along the x axis and y is the distance moved along the y axis. Of course, objects rarely move in this perculiar fashion, it's just a way of thinking about how a position is represented in Cartesian coordinates. The two parts, x and y, are referred as "components."

Cartesian coordinates

In polar coordinates there are also two orthogonal (perpendicular) axes, the radial and the tangential. The difference is that the polar axes are determined by the point itself. Imagine starting at the origin and moving directly toward the point -- this is the radial direction. The destination point is identified as (r,q), where r is the distance along the radial direction and q is the angle between the fixed axis and the radial direction, measured from the fixed axis to the radial axis in a counter-clockwise fashion. The tangential axis is perpendicular to the radial axis and points in the direction of increasing q.

Polar Coordinates

Projectile motion results when an object is subject to a single force: the constant force of gravity. In this case it is convenient to choose a Cartesian coordinate system with the y axis in the vertical direction (along a line pointing towards the center of the earth) and the x (horizontal) axis perpendicular to the y axis. If one position is represented by a pair of numbers, then the motion, which is a collection of positions, can be represented by a pair of functions

The orientation of the coordinate system was selected so that the acceleration of the object is solely in one direction, the y direction. Consequently, there is no acceleration in the x or horizontal motion, and the x motion is described by the constant velocity equation

x(t) = x0 + v0x t

where x0 is the horizontal component of the initial position and v0x is the horizontal component of the initial velocity.

There is a constant acceleration in the vertical direction, and so the vertical motion is described by the constant acceleration equation

y(t) = y0 + v0y t - (1/2) g t2

where y0 is the vertical component of the initial position, v0y is the vertical component of the initial velocity and g=9.8 m/s2, the constant acceleration due to gravity. The minus sign in the equation above is a consequence of implicitly selecting the positive y axis in the upward direction.


Part I: Vertical Launch

Let us begin our measurements with a one-dimensional, purely vertical motion.


Part II: Horizontal Launch

Projectile Data: Horizontal Shoot
Height
h
(     )
Time of
flight
(     )
Range x
trial 1
(     )
Range x
trial 2
(     )
Range x
trial 3
(     )
Range x
trial 4
(     )
Range x
trial 5
(     )
Average
Range
(     )
Stand.
Dev.
(     )
Average
Velocity
(     )
                   
                   
                   
                   
                   

Analysis.

  1. For a given height h, use g=9.8 m/s2 for the acceleration due to gravity to calculate the time the ball took in going from the launch position to the floor. Assuming there is no air resistance, the above y equation becomes

    0 = h - g t2 / 2



  2. Using this time and the average x, calculate the average initial velocity.

    xav = vav t

    You should present these results along with your data in the table above.

  3. Calculate the spread in velocities as follows. For a given h, let us take the spread in x (represented by Dx below) to be given by the standard deviation (stdev) of the x's. We could find the spread in the corresponding velocity by calculating each individual velocities and then taking the standard deviation thereof. Or we could apply a bit of calculus

    v = ( xav ± D x ) / t = vav ± D x / t,

    D v = D x / t.

    This is a simple example of what is called propagation of errors. Calculate the spread in velocities

    Propagation of Errors
    Height
    (     )
    Time
    (     )
    Spread in range
    (     )
    Spread in velocity
    (     )
           
           
           
           
           


  4. Repeat the calculation of the average speed vav and its spread for the other heights. The speed of the balls out of the launcher should not depend on the heights of the launcher. Is this what you find? Compare the vav with the velocity from Part I. (Why compare these velocities?)

    Velocity Comparison
    Velocity Part I
    (     )
    Average
    Velocity Part 2
    (     )
    % difference
         



  5. Plot h versus (average) x and fit to a power law. Compare the power you find to the theoretical prediction.

    h = (g / 2 v2) x2;

    The predicted power is square (2).



Part III: Launch at an angle

  1. Set up the launcher at a 20° angle with the horizontal such that the bottom of the launch position (circle drawn on end of launcher) is at table-top height, as shown below

    Launcher at angle



  2. Lay out the paper and carbon paper similar to Part II.

  3. Shoot the ball and measure the "range" R, the distance from the launch position to landing position. Repeat twice more.

  4. Repeat the above range measurements for 25°, 30°, ..., 65°, and 70°. Make sure that as you change the angle that the bottom of the launch position (circle drawn on end of launcher) remains at table-top height.

Analysis

  1. Make a table of your experimental ranges as well as the theoretical ranges.

    Projectile Data: Launch from an angle
    Angle
    (degrees)
    Range
    trial 1
    (     )
    Range
    trial 2
    (     )
    Range
    trial 3
    (     )
    Range
    (     )
    Theoretical
    Range
    (using Part I v)
    (     )
    Theoretical
    Range
    (using Part II v)
    (     )
    20°            
    25°            
    30°            
    35°            
    40°            
    45°            
    50°            
    55°            
    60°            
    65°            
    70°            

    The theoretical range Rth is given by

    Rth = v02 sin ( 2 q ) / g

    Use the initial velocity found in Part I and then the average initial velocity used in Part II. Which gave results closer to those in Part III?



  2. For what angle is the theoretical range a maximum? Do your measurements agree with this prediction?

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