A projectile is fired from the origin with angle of elevation ? and initial spee

Question

A projectile is fired from the origin with angle of elevation ? and initial speed v0. Assuming that the air resistance is negligible and that the only force acting on the projectile is gravity, g, we saw (through an example) that the position vector of the projectile is

?r(t) = (v0 cos ?)t?? + [(v0 sin ?)t ? 1/2gt^2 ]  ?J

(a) At what angle should the projectile be fired to achieve maximum horizontal displace- ment and what is the maximum horizontal displacement?

(b) At what angle should the projectile be fired to achieve maximum height and what is the maximum height?

Explanation / Answer

a)Let t be the time of flight.

x=v0Cos(theta)*t

t=2v0Sin(theta)/g

x=2v0^2Sin(theta)Cos(theta)/g=v0^2Sin(2theta)/g

For a constant velocity, this clearly depends only on theta and takes a maximum value of v0^2/g at theta=45 degrees.

b)For maximum height, the angle is 90 degrees or vertically.

Max. height=v0^2/2g

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