If a projectile is fired with an initial velocity v at an angle of inclination θ from the horizontal, then its trajectory, neglecting air resistance, is the parabola y = ( tan θ ) x − g 2 v 2 cos 2 θ x 2 0 < θ < π 2 (a) Suppose the projectile is fired from the base of a plane that is inclined at an angle α, α > 0, from the horizontal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by R ( θ ) = 2 v 2 cos θ sin ( θ − α ) g cos 2 α (b) Determine θ so that R is a maximum, (c) Suppose the plane is at an angle α below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R .
If a projectile is fired with an initial velocity v at an angle of inclination θ from the horizontal, then its trajectory, neglecting air resistance, is the parabola y = ( tan θ ) x − g 2 v 2 cos 2 θ x 2 0 < θ < π 2 (a) Suppose the projectile is fired from the base of a plane that is inclined at an angle α, α > 0, from the horizontal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by R ( θ ) = 2 v 2 cos θ sin ( θ − α ) g cos 2 α (b) Determine θ so that R is a maximum, (c) Suppose the plane is at an angle α below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R .
If a projectile is fired with an initial velocity v at an angle of inclination θ from the horizontal, then its trajectory, neglecting air resistance, is the parabola
y
=
(
tan
θ
)
x
−
g
2
v
2
cos
2
θ
x
2
0
<
θ
<
π
2
(a) Suppose the projectile is fired from the base of a plane that is inclined at an angle α, α > 0, from the horizontal, as shown in the figure. Show that the range of the projectile, measured up the slope, is given by
R
(
θ
)
=
2
v
2
cos
θ
sin
(
θ
−
α
)
g
cos
2
α
(b) Determine θ so that R is a maximum,
(c) Suppose the plane is at an angle α below the horizontal. Determine the range R in this case, and determine the angle at which the projectile should be fired to maximize R.
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