An object Is propelled upward at an angle θ, 45° < θ<90°, to the horizontal with an initial velocity of (Vo) feet per second from the base of a plane that makes an angle of 45° with the horizontal. If air resistance is ignored, the distance R it travels up the inclined plane is given by

Question

R = Vo^2√2
------------------ cos θ (sin θ – cos θ)
16

Show that

R = Vo^2√2
------------------ (sin 2θ – cos 2θ - 1)
32


Graph R= R(θ).
What value of θ makes R the largest? (assume Vo= 32 ft/sec.)

Answer 1

R = Vo^2√2
------------------ (sin 2θ – cos 2θ - 1)
32

R = Vo^2√2
------------------ (2sin θcos θ – cos^2 θ - 1)
32

R = Vo^2√2
------------------ (2sin θcos θ – (1-sin^2 θ) - 1)
32

R = Vo^2√2
------------------ (2sin θcos θ – (cos^2 θ - sin^2 θ) - 1)
32

R = Vo^2√2
------------------ (2sin θcos θ – (cos 2θ - 1) - 1)
32

R = Vo^2√2
------------------ (sin 2θ – cos 2θ - 1)
32

Graph R= R(θ):
The graph of R= R(θ) is a parabola with a maximum at θ = 75°. At this angle, the distance R is maximized and is equal to Vo^2√2/32.