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:
- Using Cartesian coordinates
- Using polar coordinates
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."
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.
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
- x(t): the horizontal component of position as a function of time
- y(t): the vertical component of position as a function of time
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.
- Aim the launcher (shown below, left) vertically toward a motion sensor
that is mounted on a ring stand (shown below, right)
- Load the ball into the launcher. There are different settings, i.e.
different amounts that the internal spring can be compressed. You must make
sure you always choose the same setting (same number of click sounds).
- Set up the interface/motion sensor.
- Click Start button and then shoot the launcher.
- Click Stop after the ball has landed. Ideally, if the motion is
purely vertical, the ball should be caught by the launcher.
- Copy and paste the position versus time data into Excel. The
positions are measured relative to the sensor with down as positive direction.
To use the y equation above, the measurements need to be relative to the
launcher with up as the positive direction. Your position data should have a
"flat" portion and a "parabolic" portion. To make the necessary reference and
orientation adjustments, insert a column between the time and position column.
In the inserted column subtract each position measurement from the flat
position measurement. (Explain why this subtraction method places the origin
at the launcher).
- Extract the parabolic part and fit it to a second-order polynomial (use
only the time and your inserted columns, not the original position column).
- Take a derivative of that polynomial to find the velocity as a function of
time.
- Find the velocity at the beginning of the parabolic part. (Hint:
substitute in the appropriate t, it may not be t=0.)
Part II: Horizontal Launch
- Your launcher may already have attached the bracket necessary to mount it to a ringstand.
If not, remove the nut from the top of the bracket (shown below) and attach the bracket to the nut
that is embedded in the launcher.
- Notice the
plumb line (little string with a weight on the end) and the protractor, they
will allow us to determine the angle between the launcher and the horizontal.
- Fix the launcher to a ring stand so that it is level (zero degrees from
the horizontal), as shown below.
- Take some paper and lay it out in front of the ring stand. You may need a few sheets in
order to make sure the various landing positions can be recorded.
- At the end of the launcher, there is a circle indicating the "launch
position" (the initial position) of the projectile. Note the position on the floor or lab
table (whichever you are using as the final vertical level of the projectile motion) directly
under the center of this circle. (You could use a plumb line, though not the one that
comes attached.)
- Measure h the distance from the bottom of the launch position to
the floor.
- Load the ball into the launcher. (Remember to use the same setting –
click sounds – as in Part I).
- Shoot the ball so that it lands on the paper. Note roughly where it lands
and place a piece of carbon paper there. Make sure the ringstand hasn't moved
(there's a slight recoil from shooting and/or you might have bumped the
ringstand.) Shoot the ball again. Now if you lift the carbon paper, there
should be a mark on the paper underneath where the ball landed.
- Measure x the distance between the point directly below the launch position
and the point where the ball landed on the paper.
- Repeat the above measurement four more times.
- Change the height and repeat the measurements. Do this for a total of five
different heights. There should be at least 50 cm difference between your
highest and lowest height. (There should be a total of twenty-five
measurements – 5 heights times 5 readings per height – thus far.
The column Stand. Dev. is standard deviation, use the Excel function =stdev(range)).
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 ( ) |
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Analysis.
- 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
- 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.
- 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 ( ) |
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- 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 |
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- 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
- 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
- Lay out the paper and carbon paper similar to Part II.
- Shoot the ball and measure the "range" R, the distance from the
launch position to landing position. Repeat twice more.
- 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
- 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° |
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25° |
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30° |
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35° |
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40° |
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45° |
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50° |
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55° |
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60° |
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65° |
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70° |
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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?
- For what angle is the theoretical range a maximum? Do your measurements
agree with this prediction?