Projectile Motion 5.0
Projectile Motion 5.0
Projectile Motion 5.0
________________
Projectile Motion
So far you have focused on motion in one dimension: x(t). In this lab, you will study motion in
two dimensions: x(t) , y(t). This 2D motion, called projectile motion, consists of a ball
projected with an initial velocity in the earths gravitational field.
Basic Principles
Consider launching a ball with an initial velocity vo near the surface of the earth where the
acceleration of gravity is g .
(x , y)
vo
o
yo
0
x
The position of the ball is given by the coordinates (x , y). The position of the ball depends on
time t. The motion of the ball is defined by the motion functions: x(t) , y(t). Note that at time
t = 0 , the ball is launched from the point (x , y) = (0 , yo) with the velocity vo . The initial
velocity vector vo has magnitude vo and direction o . Given the position (and velocity) of the
ball at time zero, can we figure out the position of the ball for all future time? In short, here is
the fundamental Kinematic Quest of projectile motion:
Given: The Initial Conditions: yo , vo , o .
Find: The Motion Functions: x(t), y(t) .
The quest to find how x(t) and y(t) depend on t is greatly simplified by the following facts, first
discovered by Galileo:
The horizontal x(t) and vertical y(t) motions are completely independent of each other.
x(t) = constant-velocity motion.
The theory of projectile motion goes as follows. The general motion functions for any kind of
uniformly-accelerated motion in two dimensions are
x(t) = xo + vox t + ax t2 .
y(t) = yo + voy t + ay t2 .
Projectile motion is a special case of uniformly-accelerated motion. Near the surface of the
earth, the acceleration of gravity points downward and has magnitude 9.8 m/s2 and therefore
(ax , ay ) = (0 , 9.8 m/s2). Substituting these special earth gravity values of ax and ay into the
general motion functions x(t) and y(t) displayed above, and also setting xo equal to zero for
convenience, gives the following motion functions for any object projected in the earths
gravitational field:
The Projectile Motion Equations
x(t) = vox t
y(t) = yo + voy t 4.9 t
These equations tell you everything about the motion of a projectile (neglecting air resistance).
If you know the conditions (yo , vox , voy ) at t = 0 , then these equations tell you the position
(x(t) , y(t)) of the projectile for all future time t > 0. Make sure you understand The Projectile
Motion Equations. They will be used in all future parts of this lab.
Note: In terms of the initial launch angle o , the components (vox , voy) of the initial velocity
vector vo are vox = vo cos o and voy = vo sin o .
vo
vo sin o
o
vo cos o
Exercise
The initial (t = 0) launch parameters of a projectile are yo = 3.6 m, vo = 8.9 m/s, o = 54o. Where
is the projectile at time t = 1.2 seconds ?
x = ______________ m.
y = ____________ m.
r o + vo t + g t 2 .
Note that this vector equation expresses the actual displacement r(t) ro of the projectile, as it
moves from ro to r(t) during the time t, as a combination (vector sum) of two virtual
displacements: a constant-velocity displacement vo t combined with a constant-acceleration
displacement gt2.
vo t
g t2
r(t) ro
With no gravity, the projectile would move along the tangent straight-line path at the constant
velocity vo by virtue of its inertia alone and cover the distance vot in the time t. But because
of gravity, the projectile continually falls beneath this imaginary inertial line with the
acceleration g and covers the vertical distance gt2 in the same time t.
Inertial
Motion
Falling
Motion
44.1 m
19.6 m
4.9 m
1 sec
0 sec
2 sec
3 sec
Projectile
Motion
The picture below shows the position of the projected ball at five different times. Mark the
position of the dropped coin at the same five times on the dashed vertical axis. Mark the position
of the projected ball if there were no gravity at the same five times on the dashed horizontal
axis. A ruler will help.
Projected Ball
Dropped
Coin
Floor
Exercise
The horizontal (inertial) and vertical (falling) displacements of the ball during a certain time
interval are pictured below. How fast was the ball moving when it left the edge of the table
at t = 0?
t=0
0.73 m
0.18 m
time t
vo = ________________ m/s .
Measure the distance from the center of the ball at its launch point (pictured on the side of the
launcher) to the center of the ball at its maximum-height point ... or equivalently from the
top of the ball at the launch point to the bottom of the plate.
plate
H
vo
Maximum Height:
H =
________________ m.
From this measured value of H, compute the initial speed vo of the ball. Hint: Use one of the
kinematic equations for uniformly-accelerated motion the one that does not contain the time
variable. Note that the final speed of the ball at its maximum height is equal to zero. Show your
calculation.
Initial Speed:
vo =
_________________ m/s.
Table
yo =
________________ m .
vo =
________________ m/s .
These initial parameters, which specify how you start the projectile motion, uniquely determine
the shape of the projectile path.
m/s) t
y(t) =
m) (4.9 m/s2)
t2 .
Find x(t) at t = 0 , 0.05 s , 0.10 s , 0.15 s , 0.20 s and mark these five values of x by placing five
dots at the appropriate locations on the x(m) axis shown below. Find y(t) at t = 0 , 0.05 s , 0.10 s,
0.15 s , 0.20 s and mark these five values of y on the y(m) axis below.
y (m)
0.3
0.2
0.1
x (m)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Place five dots within the xy plane pictured above that represent the actual five positions of the
projectile at the five times considered above. Label each point with the corresponding time.
Use these five points as a guide to draw the entire smooth path of the projectile as it flies through
the air.
Find the equation y(x) that describes this curved path. Hint: Eliminate t from your x(t) and y(t)
equations by solving the x equation for t and then substituting this t expression into the y
equation. You will be left with y as a function of x which should be a quadratic function of
the form y(x) = Ax2 + C. Show your derivation of y(x) in the space below. The underlined
coefficients that you fill in for the parabola equation y(x) are the special parabolic parameters
of your projectile motion.
y(x)
_____________
2
_____________ x .
Landing
Point
0
x= L
Use your parabola equation y(x) to compute the horizontal landing distance L of your projectile.
Show your calculation.
L (theory) =
_______________ m .
Launch the ball five times. Arrange for the ball to land on a piece of carbon paper, which is
placed on top of copy paper taped to the table. The scatter of landing points (dots) recorded on
the paper provides a nice visual display of the uncertainty in L. Find the average value of L and
the uncertainty in L (half-width spread around the average).
L (m)
L (experiment) =
______________ ______________ m .
Does your value of L (theory) fall within the range of the values of L (experiment)?
time
Launch
time tf
Land
Theory
The Motion Functions
Find the motion functions x(t) and y(t) that describe the motion of the ball projected at an angle
of o = 60o from your launcher. Use your measured values of yo and vo as the initial parameters
in the motion equations. Remember: vox = vocoso , voy = vosino . So as not to clutter the
equations, do not include the units of the numbers that you write in the blanks below. But make
sure that all your numbers are expressed in the metric units of meters and seconds.
x(t) =
________ t
y(t) = ________
________ t
4.9
t2 .
The Parabola
Use your motion functions x(t) and y(t) to compute the position (x, y) of the ball every tenth of a
second (t = 0.0 , 0.1 , 0.2 , 0.3 , ...) until the ball hits the table. Note that the ball hits the table
when y = 0. Record the position coordinates (x, y) in the data table below.
x (m)
y (m)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
The x and y columns of your data table describe the parabolic path of the ball through the air.
Use the program Graphical Analysis (on the Desktop) to graph y versus x. Include at least one
negative value of y on your graph. Fit the points (x, y) on your graph with a quadratic
(parabolic) function: y = Ax2 +Bx + C. PRINT the graph showing the parabolic fit. Report the
equation of your parabola:
y =
________ x
________ x
________ .
Finding R , H , tf
Find the horizontal range R and the maximum height H of the parabolic path directly on your
printed graph. DO NOT solve any equations or use . Simply look at your graph of the parabola.
Draw the horizontal line representing the table surface (the x-axis) on your graph. Write Table
Surface on this line. Mark the value of R on this x-axis. Mark the value of H on the y-axis.
Report your values of R and H here.
Note: In locating R on your graph, DO NOT go below the table surface into the region y < 0.
You can only read numbers on the x-axis and y-axis to two (or three) significant figures.
R (theory) =
______________ m.
H (theory) =
______________ m.
Use your x(t) equation and value of R to calculate the time of flight tf . Hint: At the time t = tf ,
the ball has traveled a horizontal distance x = R . Show your calculation.
tf (theory) = ______________ s .
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Experiment
Set the launcher for a o = 60o launch. Launch the ball and measure tf , H , and R. One person
should measure tf the hang timeof the ball through the air with a stopwatch. Another
person should measure H the maximum altitude with a meter stick and a simple visible
inspection of where the trajectory of the ball peaks. Use carbon paper on top of copy paper to
record the landing point. Measure the range R the horizontal distance between the launch point
and the landing point with a meter stick.
Repeat this 60o launch four more times. Fill in the table. Find the average values of your
measured parameters R , H , and tf .
H (exp)
tf (exp)
and
R (exp) = __________
is
_________ % .
and
H (exp) = __________
is
_________ % .
and
tf (exp) = __________
is
_________ % .
If any percent difference is greater than 10%, then consult your instructor.
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_______ coso t
Substitute the specified value of t = tf = 0.50 s into x(t) and y(t). Remember: At the time t = tf ,
the ball is at the landing point (x , y) = (R , 0). Solve your two equations for the two unknowns
o and R. Show all your algebra in the space below.
o =
____________
R = ____________ m.
Ask your instructor to visit your table with the net in order to check your theory and run the
experiment with your team.
The Experiment
Set the launch angle of the cannon at your theoretical (predicted) value of o. Place the landing
net on the table at your theoretical (predicted) value of R. Note that the radius of the net defines
the allowed uncertainty in R. Launch the ball. Measure the time of flight. Observe where the
ball lands. If the ball does not land in the net (cup), then move the net to the actual landing point
and try the launch again.
Actual
tf =
% diff between
___________ s .
Specified tf =
0.50 s
Actual R =
and
and
____________ m .
Actual tf = _________ s
is
______ %.
Actual R = _________ m
is
______ %.
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