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LAB III, PROJECTILE MOruO.^\I
45
Lab (^) III
Projectile (^) Motion
1 Introduction
In this lab we will look at the motion of a projectile in two dimensions. When
one
talks about a 'projectile', the implicabion is itrai we give an object an
initial velocity,
after which it moves only under the influence of graviiy, much like
a ball being thrown.
While the overall motion of a projectile may look complicated, projectile motion
can be broken (^) up into (^) accelerated (^) motion in (^) the vertical direction ("freefall, as in Lab II) and constant velocity (^) motion (^) in (^) a horizontal (^) direction.
The fundamental basis of projectile motion is that we have a constant
acceleration
in a particular direction ('downwards'). Whenever there is a constant
(^) acceleration,
we always have the freedom to choose coordinates so that one of our
axes (trre y-axis,
for example) is in the direction of the acceleration. As a result,
the acceleration is
only along one axis, with zero acceleration (constant velocity) along
the other (^) axes.
We can then decompose the motion of the projectile into horizontal and vertical
components, (^) and these components can be treated independently of one (^) another.
This technique of decomposing mo-tion into components that can be
treated (^) sep-
arately and independently is a powerful one, and greatly simplifies
many problems involving (^) motion.
2 Theory
First we choose our r-axis to be horizontal, and our
upward (see (^) Fig. (^) III.1). (^) Gravity will give (^) our (^) object
y-axis to be vertical pointing
an acceleration g downward
46 LAB III. PROJECTILE
MOTIOAI
(along the negative gr-axis), while the acceleration along the r-axis is zero:
ar :^0 (III.1a) ay :^ -g (III.1b)
where g is the acceleration of gravity. Gravity is the only force we,ll consider
in this
situation; the effect of air drag is small and (mostly) negligible.
Figure III.1: Projectile motion trajectory, where the dots are positions of the projectile
at equal time intervals.
As a result, we can write equations of motion:
n(t) _^ rs*u6at
a&) :^ !s*ustt-!rnr'
where rs and ls a,rethe values of r and y at t : l, uo, and, u6,
/ :^ 0, and g is the acceleration of gravity.
Taking the time (^) derivatives (^) of Eqn. rrr.2, (^) we get velocities:
u"(t) ar(t)
of, expressed as a vector:
u*i (^) * orj 't)ori+ (^) (ror- gt)j.
(III.2u)
(III"2b)
are the velocities at
(III.3u) (rrr.3b)
(III. au)
(III.4b)
v(f)
4B
Keep these coefficients in
example) if one measures of r and obtains:
LAB III. (^) PROJECTILE MOTIOAI
mind when fitting the trajectory of a projectile, since (for
a@) and^ fits^ the^ measurements^ with^ a quadratic function
(III.9)
Compare the terms how the fit of the
a@) :^ (0.3^ m)^ +^ (1.2)r^ - (2.1^-t)r'
it tells us that 9o :^ 0.3 m, us,sf us,: L.2 and gl(2us") :^ 2.1m-1.
in Eqn. III.8 (with ro :^ 0) with the terms in Eqn. III.9 to see
trajectory gives us values for the coefficients in Eqn. III.8.
I I t I
f r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r F 3
LAB III. PROJECTILE MOTIOAT 49
Name: Sec /Group:
Date: Pre-Lab
- Use Eqns. III.2 (^) to derive the equation for
Eqn. fII.8.
the trajectory of the projectile,
- Derive the equation for the horizontal range R for a projectile to return to the
height at which it was launched (see (^) Fig. III.1), from the initial velocity, (^) angle,
and the acceleration of gravity. Calculate the range of a projectile launched
with lvl :^ 4.7 m/s and 0 :60"^ from horizontal.
- Projectile Motion Simulation
See http : / /wvw.physics. drexel. edu/labs/ for simulation.
(a) Pick an angle (d) and velocity (os) for the simulation so the range is about
1 m and the height is about 0.6 m. Enter the values you used in Table. III.1.
LAB III.
Name:
PROJECTILE A/IOTIOAI
Sec (^) /Group: Date:
Table (^) rrr.l: (^) PRE-LAB projectile (^) Motion (^) simulation (^) Data
l.ro (^) I
r motion
51
Lr
Lt nt usN A'r
nor _^ Ito (^) I cos I
y motion y(t)
Fit
t
t
t
[*/']
[*/r']
Trajectory a@)
Fit
t
t t' L
[*/']
[d.s]
[-]
l'l
[-/']
[*/']
_ (^) cocftczt
Meanirg of Coefficient
l
J^ I
l
Cg:
C1 :
C2:
u0y
0,y
d,g :
dt:
d2:
Meaning of Coefficient
Calculated:
Range l] Measured:^ fl
<>
.>
LAB III. PROJECTILE MOTIOAT 53
4 Equipment
Figure III.2 shows the launcher that you will use for this lab. Note that the base of
the launcher needs to be held securely (preferably clamped to a lab bench) when a
ball is launched, so that direction of the launcher is unchanged when the trigger-cord
is pulled. CAUTION: keep clear of the barrel of the launcher when it is
loaded, and^ make sure that the area tdownranget^ is clear. Warn anyone
nearby to watch out and keep clear before firing the launcher. Remember
that the ball may (^) bounce off the floor, wall, or other (^) obstacles and go further than anticipated.
Figure III.2: Projectile launcher
W}
't
Figure III.3^ shows a close-up
sure that the thread is free to
accurate angle.
of the angle setting
move,, and let the
for the projectile hanging weight
launcher. Make
settle to get an
5 Procedure
5.1 Altitude, Velocity and (^) Range
- Adjust the launcher so that it (^) is pointing (^) verticallr (^) @: 90"), and place (^) it on the floor near (^) a wall.
2. Starting from the height of the launcher, hold a meter stick against the wall.
3. Insert a^ ball into the launcher, pushing^ it only to the first 'stop' (lowest velocity
setting). Keep clear of the launcher opening when it (^) is loaded!
56
fI
,>\ - T
I I
5 il I rt-t I I",I I !t -T ;l I ;, I F F F 7 7 7 F A F
F^ /
- F F F ; /-- F F F tl ,; F .- I ; I T ; i i I
LAB III. PROJECTILE MOTION
plot your y positions a,s a function of^ time,^ and^ fit^ with^ a^ quadratic^ (polynomial
of degree:2), and^ enter^ the^ fit^ coefficients^ (y6'^ ?ss^ a'nd^ -gl2)^
in Table III'4;
turn in your plot with^ your^ lab report'
Plot your^ g positions^ as^ a^ function^ of^ r^ position^ and^
fit with a quadratic'^ Enter
the fit coefficients^ in^ Table^ III'4'
I r r r r r f f r f r f f r r r r r r r r r r r r r r r r r r r fr fr r r- LAB III. Name:
PROJECTILE MOTIOAI
Sec /Group:
I)ate: Table fff.2: Launch velocity (^) and range launcher setting (^) height (^) [] us (^) [/r] 1
1r^ )
3 fi(calc) for 0-60' B(meas) for 0- (^) 60" b( Data [] []
.-.
,L
Jt
,ll}.
A
-. A
.>
-,
^>.
,>.
tL t
tl. rL
tr
t
l.
lL
'
D
D,
1>.
'L tr\
i>
1}' L.
L L L L I I
LAB III"
Narne:
PROJECTILE MOTlOr\r
Sec /Group:
--
I)ate:
Table III.4: Projectile motion results summary.
[*/']
[-]
[*/'] %
[*]
59
r as. t^ slope (ro")'
r us. t intercept (ro)'
lroIcos 0 - ?)or^ (calc): uor (meas-calc) %ditrerence:
a(t)- co^ *^ cfi^ *^ c2t
cs (yo)'
c1 (ror)t
c2 (-^ s l2),
I - -2c2:
a@) (^) -dodfid,2r da:
dt:
dz:
[*/']
[*/t']
[r"/t']
t
How does (^) the measured range compared to
the percentage^ difference. If you find that
some possible causes for the difference.
7 Conclusions l-, fr f^ r^ f f^ f^ f^ J'-^ f^ r^ r^ f r f f f, f f f. f f f f r f f r f f F-f f fr
LAB III. PROJECTILE MOTIOAI
Name: (^) Sec (^) /Group: Date: the range you predicted? Determine the difference is significant, suggest 61
Use the results of the
of (^) A us. r should (^) be, fits for r ns. t and y
and cornpare to the
us. t to calculate what the coefficients coefficients from vour fit.
- Does^ your^ data show acceleration along the
on which you base your conclusion)? What
acceleration along r?
r axis (indicate the data or plot
effects could^ cause vou to see an
- How^ does^ the^ vertical^ acceleration in your
'standard' value^ of^ 9.80^ mf^ s2?^ Calculate the
measurements compare with the
percentage difference.
