Learn about projectile movement by
running and jumping through
track and field events!
Distance and Displacement
In physics, we can measure an object's
change in position as distance or
displacement. While distance calculates
the length of the full path the object
took between the two points, distance
only measures how far the two points
are from each other (that is, the closest
path between them).
Suppose a runner runs around a
oval track that measures 400 meters around,
and ends up at the same position he started at.
Then the distance between his start and
end point is 400 meters, but the displacement
is 0. Because he ended up at the same
point, displacement measures as if he
hasn't moved at all!
Similarly, if you jump up (vertically) 10 centimeters
and then land exactly where you jumped from,
the distance from your starting position to
your ending position is 20 centimeters
(10 centimeters going up, 10 centimeters going back down).
But your total displacement is 0 again.
Finally, suppose
(without moving "back" or "forward"
in the direction of the front of the podium)
an athlete steps from
flat ground, 2 feet away, up onto the
center of
a 2nd
place podium that is 4 feet wide
and 3 feet tall.
Horizontally, you have to move 2 feet
to get to the podium, then 2 feet to the
center (so 4 feet total). Vertically, you have to move
3 feet.
So, the displacement between the starting and
ending points is 4 feet + 3 feet, a total of 7 feet.
But (by the Pythagorean Theorem)
the distance between your starting and
ending points is 5 feet.
Displacement and distance on a podium
Velocity
Velocity refers to the rate at which an object
moves in some direction.
For instance, if you run straight on a track
at 5 meters per second (written as "5 m/s"), then
each second, you will have moved another 5 meters.
The table below shows the total distance
you have traveled (in meters) for the first
three seconds that you are running.
Time (s) from start
Distance (m) traveled
0
0
1
5
2
10
3
15
So, suppose you run 15 meters in 3 seconds
(like the table suggests).
To determine your average velocity,
you can calculate overall distance divided
by overall time:
15 m
= 5 m/s
3 s
While velocity measures a change in position
over time, acceleration measures
the change
in velocity over time (that is, how much
does the object speed up or slow down).
Vertical Velocity
The previous section concerned "horizontal velocity",
like running. To simplify calculations (generally
without significant error) we assume that this is happening on
a flat plane, rather than say, over a small hill.
To look at "vertical velocity", we consider movement
away or towards a flat plane, like jumping up and down
in one spot on a hardwood floor. A major difference
is when jumping, we have gravity pushing us in the
opposite direction.
This downward force determines how quickly our
velocity changes. The force (called the standard acceleration
of gravity or acceleration due to gravity
or sometimes even shortened to just gravity)
is written as g and is around
9.8 m/s2.
(Note: this value is the acceleration
near the Earth's surface,
but for our calculations in this game,
it makes sense to assume the moving object
is near the Earth.)
Projectiles
A projectile is an object that is sent
through the air along
a curved path under the action of gravity only.
It can also be anything thrown, kicked, etc., but
for our game, a person jumping is the projectile.
We will also simplify things by focusing on
one point of the jumping object- specifically,
the center of the person's feet.
(In real track
and field calculations, calculations would focus
more on the person's "center of gravity").
A projectile's general movement can be calculated based
on its horizontal velocity and vertical velocity,
as we did in the table earlier - however, in this case we
must remember that gravity is constantly affecting
the vertical velocity. Let's start with the same table,
and add an initial vertical velocity of 9.8 m/s.
To visualize this as a graph, we'll rewrite
distance traveled as x (for horizontal displacement,
in meters) and y (for vertical displacement, in meters).
Time (in seconds) from start is abbreviated to just t.
t
x
y
0
0
0
1
5
9.8
2
10
9.8
3
15
0
Notice a few things: even though the vertical
velocity changes during flight, the horizontal
velocity (of 5 m/s) stays the same throughout.
And, the rising vertical values match the descending
vertical values, so an arc graphing the
x and y
values would be horizontally symmetric.
Horizontal symmetry of a projectile path
Formulas
The formulas below assume no air resistance,
and assume that objects begin and end the
projectile path on the same flat plane.
(Note: by "jump velocity" or v0 we mean the
velocity at which an object is initially moving
along horizontal and vertical directions together;
i.e., basically the "diagonal" velocity of
a running jump. We use θ to denote the angle of the
diagonal launch.)
Horizontal and Vertical from Jump velocity:
Initial horizontal velocity = v0
cos(θ)
Initial vertical velocity = v0
sin(θ)
Total time that the projectile is in the air:
2 v0
sin(θ) / g
Distance of arc:
v0
cos(θ) × (total time in the air)
Maximum height of arc:
v0
sin(θ) × (half the total time in the air)
- .5 × g × (half the total time in the air)2
Note: your own answers may not exactly match
those in this game
due to rounding performed in calculations.
