PASCO ME-6853 Shoot-the-target Accessory - Manual (Page 2)

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S h o o t- t h e - t a r g e t A c c e s s o r y T h e or y
2
D
h
q
V0
Launcher
Line of sight
Target
Trajectory
Projectile
Before it falls, the target is attached to the drop box by a permanent magnet so it can
hang indefinitely, even when the drop box is not powered. A photogate detects the
projectile as it leaves the launcher and triggers the drop box. The drop box releases
the target by driving a current through a coil that cancels the field of the permanent
magnet.
Theory
The target’s initial velocity is zero, and if falls
straight down with constant downward accel-
eration. The initial velocity of the projec-
tile as it leaves the launcher has both
horizontal and vertical compo-
nents, but it undergoes the
same acceleration as the target.
At elapsed time t, the vertical position of the target is given by
(eq. 1)
where h is the initial height of the center of the target (measured from the elevation of
the end of the launcher), and g is the gravitational field strength.
The vertical and horizontal positions of the projectile are given by
(eq. 2)
and
(eq. 3)
where V0 is the launch speed of the projectile as it leaves the launcher.
If the horizontal distance from the launcher to the target is D, and the projectile will
reach the target’s horizontal position at time t', then
(eq. 4)
By examining the diagram, we can also see that
(eq. 5)
Combining equations 4 and 5 and using the trigonometric identity
tan(
θ) = sin(θ)/cos(θ) yields
(eq. 6)
y
T
t ( ) h
1
2
---gt
2
– =
y
P
t
( )
V0
θ ( )t
sin
1
2
---gt
2
– =
x
P
t
( )
V0
θ ( )t
cos =
D V0
θ ( )t
cos
′
=
D
h
θ ( )
tan
---------------- =
V0
θ ( )
sin t
′
h =