This exercise analyses motion in which the acceleration is (almost) constant.
The vertical acceleration is largely due to gravity (g downwards), and
horizontal accelerations are relatively small. With these approximations,
we may use the standard expressions for velocity v and displacement in
the vertical (y) and horizontal (x) directions as functions of time t:
where subscript 0 identifies initial values. These equations
are derived in Appendices 1 and 2—check that you are comfortable
with using them. If we substitute from (4) into (2) to eliminate t, we
obtain y(x), that is to say the path of the motion. Appendix 3 shows that
this is a parabola.
But first let's ask: how well do the simple projectile equations
work? Are the motions in the x and y directions independent?
How constant is vx? How linear is the decrease in vy with time?
Homework Question 1: background.
When you aim a gun, the light travels in a straight line from the
target to your eye. If you fired a laser along that line (the aiming
line), the laser light would hit the target. However, when you fire
a bullet along that line, it does not travel in a straight line,
because it is affected by gravity.
How far does a projectile fall below the aiming line? This is shown
schematically in the figure at left: an object is projected in the
direction of the aiming line, but falls further and further below
it as it moves to the right. We describe this mathematically in
Appendix 3, but let's think about it first. The sequential positions
of the bullet shown at left are equally spaced in the x direction,
2 units apart. If vx is constant, then these represent equally spaced
times. The vertical lines between the aiming line and the trajectory
represent how far the bullet has fallen below the aiming line. Setting
vx0 and y0 to zero in (2) shows that the distance fallen by an object
is proportional to the square of the time. So these vertical lines
should have lengths that increase in the ratio 0, 1, 4, 9...n2.
In Appendix 3, we show that the length of these lines is
From (2), Dy
is also how far an object falls in time t, starting from rest. This
result is immortalised in an ancient physics problem called The
Monkey and the Hunter. (In case you are worried, we have never
heard of a physicist who hunted real monkeys).
1In
school you may have used the notation
and so may wonder why we use a different notation. First, for motion
in two dimensions, we need to separate components, hence the subscripts
x and y. Second, it is not always possible or convenient to start
the object at (x,y) = (0,0). If two start from different places, at
least one must have different initial coordinates. There may be other
reasons to chose the origin independently of the projectile.
The Monkey
and the Hunter
Download Print Version for
Homework Question 1 (625k pdf file)
A monkey hangs from the branch. A hunter
aims a rifle at him and fires. At the instant that the gun fires,
the monkey sees the smoke and immediately lets go the branch and starts
to fall. Is this a good idea for the monkey? Fig 1 may be used to
predict what happens—if the motion in the x and y directions
are really independent. Which is what we shall check.
Your teacher will show you a demonstration of this situation. However,
it all happens very quickly. For this reason, a high speed video (in
the making of which no monkeys were injured) is available above. (also
see www.phys.unsw.edu.au/~jw/demo/projectiles.html
for a fuller explanation)
The following questions use stills
from this video. An ordinary video camera runs at 25 frames per second.
This is of limited use for projectile motion, because an object moving
at 1 ms-1 travels 40 mm in 1/25 seconds, and so it appears
blurred. This one was taken at 800 frames per second, each exposure
lasting 1 ms. (You may wish to calculate how far the projectile travels
in 1 ms during the last frame, and use this as part of your answer
to question 1c.) The elapsed time is shown in the stills. The grid
drawn on the blackboard has lines separated by 100 mm.
Homework Question 1.
a) Using a ruler, measure the width of several squares on the
grid. Determine how to convert millimetres measured with a ruler
in the image to metres in the plane of the motion.
b) Estimate the error in your measurements of Dx
and Dy.
c) Is vx constant for the projectile? Explain your answer quantitatively,
using measurements.
d) Using a ruler, carefully measure the vertical distance Dy
between the projectile and the aiming line for each image. Equation
5 predicts that Dy = ½gt2.
What sort of a plot of the variables2 Dy
and t would give a straight line?
e) Plot the line suggested in (d). Within the precision of your
measurements, does the projectile fall a distance proportional
to t2?
f) Draw a line through your graph and use it to calculate g.
What value do you get, and what is the estimated error? Comment.
2Why
do we plot the data so that a theory gives a straight line? Well,
it is easy to recognise whether a line is straight or not. It is much
harder to recognise whether a curve is ,
because this curve looks qualitatively similar to .
Experimental Exercise
Equipment: projectile launcher, ball, plumb bob,
rulers, graph paper, carbon paper, tape. Safety note: The projectile launcher is not heavy,
but we don't want it falling on your head while you are on your
hands and knees searching for a ball that got away. Take care.
The projectile launcher is sketched below. A ball is released at
height H. Select a convenient H, mark it and then keep it constant.
At the end of the track, the projectile is travelling horizontally.
a) Calibration. For this stage, the ball
should land on the table. Keep the launching end well away from
the table so that balls do not fall on the floor. Ensure that
it is horizontal so that the projectile will be launched horizontally.
Measure the height h. For safety reasons and stability, one person should
hold the launcher.
Release the projectile from the chosen point and observe
where it lands. Tape the graph paper to the bench so the ball lands approximately
in the centre. Refine your technique to improve repeatability.
b) Place the carbon paper, black side down, on top of
the graph paper (no tape needed). Now you can release the ball and it
will make a mark on the graph paper. Repeat this procedure a dozen or
so times so that you have an estimate of the repeatability of the procedure.
Use the ruler to measure the distance from x=0 to the graph paper. From
the marks on the paper, obtain a value of the horizontal distance3
travelled R, and an estimate DR
of the error in R.
c) Also obtain an error Dq
in the lateral angle. (Over how many radians either side of the middle
of the distribution does the ball fall?)
d) From your measured values of h and R, calculate v0, the
launch speed of the projectile with your chosen value of H. (You may wish
to refer to Appendices 1 and 2. Hint: you can find vx0 from
the distance travelled and the time of flight t. What is t for an object
falling from height h?)
3This is often called the range. Do
not confuse this use with the range of experimental results (xmax
- xmin).
In the next part of the exercise, you
will place the projectile launcher so that its end is near the edge
of the table, as shown at left. Do not do an experiment
at this stage.
Instead, use your value of v0 and the new value of h
to calculate the new value of R required to land the ball in a cup
with height d = 100 mm. (Do not forget to include d in your calculations.)
When you have calculated R ± DR,
make sure that all team members have agreed on the calculation and
error.
Now call the demonstrator to observe while you conduct the experiment.
You will be marked on whether the ball lands in the box,
based on your calculation alone, with no rehearsals.
If not, the demonstrator will ask you to repeat both experiments,
but this time using a different value of H in both cases.
Calculation of R
and DR.
(Hint: this is the reverse of your calibration experiment. You know
vx0, you need the distance travelled in the x direction,
so you need the time of flight.)
d) Now make a prediction. If we replaced the ball with a toy car
(with smoothly turning wheels and negligible air resistance), will
it travel further or less far than the ball? Why?
What about a metal cylinder or a section of pipe?
Follow up Homework Question
A ski jump has a take-off angle of q.
A skier lands after having travelled a horizontal distance R. Her
landing point is a height h below the take-off point.
a) How fast was she travelling when she took off?
b) Before you go any further, check that your answer makes sense
by considering two special cases. First, consider q
= 0, and compare this with the expressions you used
in the exercise.
c) Second, consider the case where tanq
= -h/R. Does the value for v0 make sense
here? (One aspect of the exercise is relevant here, too.)
d) Finally, put in some reasonable values for q,
h and R and see what value you get for v0. Express your
answer in km/hr.
Optional questions. (If these
seem too hard now, return to them later after you have studied more
mechanics.)
1) Think about the relationship between vo and H. Compare
this with your results in part (c) of the experimental exercise.
Does the value you obtain surprise you? If so, where might extra
energy have gone? You might also look at your answer to (d).
2) How important is air resistance? If your cross sectional area
is A, and if you accelerated all of the air in front of you up to
the speed at which you were travelling, what force would the air
exert on you?
Appendix 1.
Are horizontal and vertical motion really independent?
One example that depends on this question is sometimes
asked thus: If you fired a bullet horizontally and simultaneously
let another bullet fall, which would reach the ground first?
If the horizontal and vertical motion are independent of each other,
then the two projectiles should hit (level, flat) ground at the
same time, as indicated by the diagram at left (Fig A1).
In this session, apparatus will be available to investigate
this.
Fig A1. The velocity components, at equal time
intervals, for two projectiles whose v0y is initially zero.
One falls vertically (v0x =0). For the other, v0x>
0 and so it follows a parabola.
Appendix 2.
Are horizontal and vertical motion really independent?
and so may wonder why we use a different notation. First, for motion
in two dimensions, we need to separate components, hence the subscripts
x and y. Second, it is not always possible or convenient to start
the object at (x,y) = (0,0). If two start from different places, at
least one must have different initial coordinates. There may be other
reasons to chose the origin independently of the projectile.
,
because this curve looks qualitatively similar to 
.
