TUTORIAL
In physics, a projectile starting at ground level at an angle θ and initial velocity v, has x- and y- coordinates that change with time t according to (air resistance negligible):
x(t)=v*cos(θ)*t
y(t)=v*sin(θ)*t-16*t2
t in seconds, x and y in feet.
step 1: click on the button above " click here to start " to start the applet. Click on the button "auto" to start the projectile. Click on the button "pause" to stop the animation. Click on "pause" again to continue. Note the time (seconds), x(t) (feet) and y(t) (feet) on the right (pink panel).
step 2: Solve x(t) for t to obtain
t = x(t)/v*cos(θ)
Substitute t in y(t) to obtain an equation of the form
y=ax2+bx
and show that
a = -16/(v*cos(θ))2
b = tan(θ)
Read values of x and their corresponding y (pink panel in applet), substitute in the equation above and check. (small differences occur due to errors of approximation).
step 3:The range is the distance between the point(0,0) and the point where the projectile touches the ground on the x-axis.
Let the speed v = 150 ft/sec and change angle θ. Make a table with different values of angle θ and the corresponding range.(the range is the x coordinate displayed in the pink panel, top right, once the projectile has touched ground).
What angle gives maximum range?
The range can be found by solving
y=ax2+bx = 0
for x.
Show that the range R is given by
R =-a/b = v2sin(2θ)/32
For what value of θ (0 < θ < 90) R is maximum?
Compare graphical (applet) values and values obtained using the above formula.
More
Mathematics Applied to Physics and Engineering.
