Projectile Motion: Can You Predict Exactly Where the Ball Will Land?
There are some physics experiments where the result is almost guaranteed to make a student stop and think.
Projectile motion is one of them.
Put a ball on the edge of a table and let it fall. There is nothing particularly surprising about that.
Now launch the same ball horizontally from the table and it travels forwards while falling towards the floor.
Again, that seems fairly obvious.
But measure the height of the table, measure the horizontal speed of the ball, calculate where physics says it should land and then put a small target at that exact position.
Suddenly it becomes much more interesting.
You are no longer simply observing motion.
You are predicting the future.
And, if the measurements are good enough, the ball lands remarkably close to where the equations say it should.
That is one of the reasons projectile motion deserves a place in a home physics laboratory.
The Big Idea: One Ball, Two Motions
The most important idea behind projectile motion is surprisingly simple:
The horizontal and vertical motions can be considered independently.
A ball projected horizontally is doing two things at the same time.
Horizontally, it is moving forwards.
Vertically, it is falling.
Ignoring air resistance, there is no significant horizontal force acting on the ball once it has been launched. Therefore its horizontal velocity remains approximately constant.
Vertically, gravity accelerates it downwards at approximately:
g = 9.81 m/s^2
So we can analyse the two directions separately and then combine them to understand the complete curved trajectory.
This is an extremely important step in learning mechanics because students begin to realise that complicated-looking motion can sometimes be broken into simpler components.
Experiment 1: Launch a Ball Horizontally
The simplest version requires surprisingly little equipment.
You need:
a small ball;
a table or raised platform;
a ruler or tape measure;
some way of launching the ball consistently;
paper or card for marking the landing position;
and ideally a method of measuring the launch speed.
A steel ball bearing works particularly well if a suitable ramp or launcher is available.
The important point is that the ball should leave the end of the apparatus horizontally.
First Measure the Height
Measure the vertical distance from the point where the ball leaves the launcher to the floor.
Suppose the height is:
h = 0.80 m
The vertical motion begins with essentially zero vertical velocity because the ball is initially travelling horizontally.
For an object falling under gravity:
s = ut + 0.5at^2
Since the initial vertical velocity is zero:
h = 0.5gt^2
Therefore:
t = sqrt(2h/g)
Putting in our example values:
t = sqrt((2 x 0.80)/9.81)
t = approximately 0.404 s
That means the ball should be in the air for about four-tenths of a second.
Notice something particularly interesting here.
The calculation contains no horizontal velocity.
The time taken to reach the floor depends on the vertical motion, not on how quickly the ball is travelling horizontally.
Now Predict the Horizontal Distance
Suppose we measure the horizontal velocity of the ball as:
v = 2.0 m/s
There is approximately no horizontal acceleration, so:
distance = speed x time
Therefore:
x = vt
x = 2.0 x 0.404
x = approximately 0.81 m
Physics therefore predicts that the ball should land about:
81 cm from the point directly underneath the launcher.
Now comes the enjoyable part.
Measure 81 cm across the floor.
Put down a piece of paper.
Draw a target.
Launch the ball.
Does it hit?
Turn the Calculation into a Prediction
I particularly like experiments where the calculation happens before the observation.
It changes the psychology of the practical.
Instead of:
"Let's do the experiment and see what happened."
the student is saying:
"Physics says the ball is going to land there."
The experiment then becomes a test of the model.
Move the target only after making the prediction.
If the ball lands near the centre, the calculation suddenly feels much more meaningful than an exercise printed in a textbook.
What If We Launch the Ball Faster?
This produces another useful investigation.
Keep the height exactly the same but change the horizontal launch velocity.
Perhaps try:
1.0 m/s
1.5 m/s
2.0 m/s
2.5 m/s
The time taken to fall should remain approximately the same.
But the faster-moving projectile travels further horizontally during that time.
For example, if the ball remains airborne for 0.40 s:
At 1.0 m/s:
x = 1.0 x 0.40 = 0.40 m
At 2.0 m/s:
x = 2.0 x 0.40 = 0.80 m
At 2.5 m/s:
x = 2.5 x 0.40 = 1.00 m
The relationship should be approximately linear.
Double the horizontal speed and, for the same height, you should approximately double the horizontal range.
That gives us another experiment.
Plot:
horizontal range
against
horizontal launch velocity.
What should the graph look like?
Physics predicts a straight line through, or very close to, the origin.
The Experiment That Often Surprises Students
There is another lovely demonstration.
Take two identical balls from the same height.
Allow one simply to fall vertically.
Launch the other horizontally at exactly the same instant.
Which reaches the floor first?
Many students instinctively predict that the stationary ball will land first because the projectile has "further to travel".
But that mixes horizontal and vertical motion together.
Their vertical motions are essentially identical.
Both balls:
start at the same height;
have zero initial vertical velocity;
experience the same gravitational acceleration.
Therefore, ignoring air resistance and assuming they are released simultaneously, they should reach the floor at almost exactly the same time.
The horizontally launched ball simply travels sideways while falling.
That is a beautifully simple demonstration of the independence of horizontal and vertical motion.
Galileo Would Recognise the Idea
This experiment also gives an opportunity to connect modern school physics with the development of scientific thinking.
Galileo's work on falling bodies and projectiles helped establish the idea that projectile motion could be understood by combining two simpler types of motion.
Horizontal motion could continue at constant velocity while vertical motion was accelerated by gravity.
Combine them and the resulting trajectory is a parabola.
Today we can demonstrate the same idea with equipment Galileo could hardly have imagined.
A smartphone camera can record hundreds of frames.
Video-analysis software can measure position frame by frame.
A computer can plot x and y coordinates separately.
Yet the physics underneath remains beautifully simple.
The More Advanced Experiment: Film the Projectile
Once the basic target experiment works, video analysis takes it much further.
Position a camera side-on to the trajectory.
Ideally:
use a tripod;
keep the camera perpendicular to the plane of motion;
put a metre rule or known scale in the image;
use a high frame rate if available;
provide good lighting;
and use a background that makes the ball easy to see.
Record the launch.
Then track the centre of the ball frame by frame.
For every frame you can obtain:
time;
horizontal position x;
vertical position y.
Now analyse the two directions separately.
Horizontal Position Against Time
Plot:
x against t
For ideal projectile motion, this should give approximately a straight line.
The gradient represents the horizontal velocity.
So:
horizontal velocity = change in x / change in t
The important observation is that the gradient should remain approximately constant.
That is evidence that the projectile has approximately constant horizontal velocity.
Vertical Position Against Time
Now look at the vertical motion.
Plotting vertical displacement against time should not produce a straight line.
The ball is accelerating.
For a horizontal launch:
y = 0.5gt^2
So the vertical displacement is proportional to t^2.
A particularly good analysis is therefore to plot:
y against t^2
That should give approximately a straight line.
Since:
y = 0.5gt^2
the gradient should be approximately:
g/2
or about:
4.9 m/s^2
Double the gradient and you have an experimental estimate of gravitational acceleration.
Suddenly one projectile experiment has become a measurement of g as well.
Plotting the Actual Trajectory
The video data can also be used to plot:
y against x
The result should be the familiar curved projectile trajectory.
For a horizontal launch:
x = vt
so:
t = x/v
Substituting this into:
y = 0.5gt^2
gives:
y = gx^2/(2v^2)
This is of the form:
y = kx^2
which is a parabola.
That familiar textbook curve is therefore not something we simply have to accept.
We can generate it from measurements made in the laboratory.
A Particularly Powerful Way to Teach It
One difficulty with mechanics is that students can become very good at selecting equations without really understanding the motion.
They may know:
s = ut + 0.5at^2
and:
v = u + at
but treat them as formulas to be searched through until one happens to contain the variables in the question.
Projectile motion forces a better way of thinking.
Before writing any equation, ask:
Which direction am I considering?
Horizontally:
a = 0
Vertically:
a = g
Then analyse each direction separately.
This simple habit solves a remarkable number of projectile-motion problems.
What About Projectiles Launched at an Angle?
Once the horizontal-launch experiment is understood, we can go further.
Suppose a ball is launched with speed u at an angle theta above the horizontal.
Its velocity can be resolved into two components.
Horizontal component:
u_x = u cos(theta)
Vertical component:
u_y = u sin(theta)
The horizontal component remains approximately constant.
The vertical component changes because of gravity.
The same basic principle still applies:
separate the motion into horizontal and vertical components.
The problem only appears more complicated because the projectile now begins with a vertical velocity as well.
Why Approximately 45 Degrees Gives Maximum Range
If a projectile is launched and lands at the same height, ignoring air resistance, its range is:
R = u^2 sin(2theta)/g
The maximum possible value of sin(2theta) is 1.
Therefore:
2theta = 90 degrees
so:
theta = 45 degrees
This gives another excellent experiment.
Keep the launch speed approximately constant and investigate launch angles such as:
20 degrees
30 degrees
40 degrees
45 degrees
50 degrees
60 degrees
70 degrees
Measure the range each time.
Does the maximum occur near 45 degrees?
Even better, compare complementary angles.
For example:
30 degrees and 60 degrees
or:
20 degrees and 70 degrees.
Ideal projectile theory predicts that complementary launch angles should produce the same range when launch and landing heights are equal.
Real experiments will not be perfect, which gives us something else to discuss.
When Reality Refuses to Behave Perfectly
A real projectile experiment rarely gives exactly the theoretical answer.
That does not make it a bad experiment.
Quite the opposite.
It gives us an opportunity to ask why.
Possible sources of discrepancy include:
air resistance;
uncertainty in measuring the launch velocity;
the ball not leaving perfectly horizontally;
variation in the launcher;
inaccurate height measurements;
difficulty identifying the first point of contact with the floor;
camera perspective;
uncertainty in locating the centre of the ball in video frames;
and rotation or spin of the projectile.
For short-range experiments using dense balls, air resistance is usually fairly small.
Measurement uncertainty is often much more significant.
That itself is an important scientific lesson.
When an experiment disagrees slightly with theory, we should not immediately declare that Newtonian mechanics has failed.
We should first examine the quality of our measurements.
Add a Target and the Experiment Becomes Much Better
A small addition transforms the whole practical.
Do not merely measure where the projectile lands.
Predict it first.
Draw concentric circles on a sheet of paper.
Mark the calculated landing position in the centre.
Perhaps award:
10 points for the centre;
5 points for the middle ring;
1 point for the outer ring.
Now repeat the calculation for different launch speeds or heights.
It becomes almost a physics version of target shooting.
For younger students in particular, this creates a memorable question:
Can your calculation tell you where to put the target?
A Further Challenge: Change the Height
Instead of changing the launch velocity, change the height.
Since:
t = sqrt(2h/g)
the flight time is proportional to sqrt(h).
The horizontal range is:
x = vt
so, for constant launch speed:
x = v sqrt(2h/g)
Therefore:
x is proportional to sqrt(h)
That gives another useful graphing investigation.
Measure the range from several different heights and see whether the mathematical relationship appears in the data.
This moves the activity beyond simply confirming one numerical prediction.
Students begin investigating relationships between variables.
Could You Calculate the Launch Speed Without Measuring It Directly?
There is another nice reversal of the experiment.
Suppose you know:
the height h;
the horizontal range x;
and g.
From:
t = sqrt(2h/g)
and:
x = vt
we obtain:
v = x/t
Therefore, by measuring how far the ball travels before hitting the floor, we can calculate its horizontal launch speed.
So projectile motion can become an indirect method of measuring velocity.
That is an excellent example of something physicists do constantly:
measure quantities that are easy to obtain and use theory to determine something that is harder to measure directly.
Safety Matters
Projectile experiments do not need to involve high speeds.
For a home laboratory or classroom investigation, a small low-energy projectile travelling a metre or two is more than enough.
Keep the trajectory away from:
faces;
windows;
computer screens;
fragile laboratory equipment;
pets;
and anybody unexpectedly walking through the experimental area.
A steel ball bearing is excellent experimentally but should be used with a suitable catch area so that it does not bounce unpredictably around the room.
There is no educational advantage in increasing the launch energy unnecessarily.
Why I Like Projectile Motion So Much
Projectile motion sits at an interesting point in physics education.
The equations are not especially difficult.
The apparatus can be remarkably simple.
Yet underneath it are several extremely important ideas:
vectors;
resolving velocity into components;
constant velocity;
acceleration;
gravitational motion;
graphical analysis;
mathematical modelling;
uncertainty;
and experimental testing.
Most importantly, students get to see that a curved trajectory does not necessarily require a complicated explanation.
Break the motion into two directions and everything becomes much clearer.
That principle extends far beyond projectiles.
It is part of the wider language of mechanics.
The Moment That Makes the Experiment Worthwhile
There is a particular moment I enjoy in this experiment.
The measurements have been made.
The equations have been used.
Someone measures across the floor and carefully puts down the target.
The ball has not yet been launched.
At that moment physics has made a prediction.
Then the ball is released.
It travels from the table, follows its curved path and lands.
If it strikes close to the predicted point, the equations suddenly cease to be abstract symbols on a page.
They have described something that actually happened.
And that, for me, is exactly what practical physics should do.
Do not just launch the projectile and measure where it lands.
Measure the system, calculate where it ought to land, put the target there — and then see whether physics gets it right.


