Problem: For a projectile lunched with an initial velocity of v0 at an angle of θ (between 0 and 90o) , a) derive the general expression for maximum height hmax and the horizontal range R. b) For what value of θ gives the highest maximum height?Solution The components of v0 are expressed as follows:vinitial-x = v0cos(θ)vinitial-y = v0sin(θ) b)The distance traveled by a projectile follows a uniform motion, meaning, velocity is constant from the start point until it reach the ground along the horizontal axis, so, the range Rcan be expressed asR= Vinitial-xSubstituting the initial velocity on the x-axis results to the followingR=(But, the time it takes a projectile to travel this distance is just twice of tmax-height by substitution, we obtain the following:R =x 2 x (Re-arranging and then applying the trigonometric identitysin(2x) = 2sin(x)cos(x)we arrive at the expression for the range RasR =sin a)Let us first find the time it takes for the projectile to reach the maximum height.Using:Vfinal-y = Vinitial-y* aysince the y-axis velocity of the projectile at the maximum height isVfinal-yThen.= Vinitial-y+ ytSubstituting the expression of vinitial-y and ay = -g, results to the following:Thus, the time to reach the maximum height isEmax-height"We will use this time to the equationYfinal - Yinitial = Vinitial-yt + (1/2)a,?if we use the time taken to reach the maximum height, therefore, the displacement will yield the maximum height, sohmax = Vinitial-yt + (1/2)a,rsubstituting, the vinitial-y expression above, results to the followinghmax =Then, substituting the time, results to the followinghmax =()+ (1/2)aySubstituting ay = -g results tohmax =()- (1/2)gsimplifying the expression, yieldshmaxX sin

Name: Problem: For a projectile lunched with an initial velocity of v0 at an
Uploaded: 2024-03-20T06:57:47.000Z
Channel: Bartleby
Description: Problem: For a projectile lunched with an initial velocity of v0 at an angle of θ (between 0 and 90o) , a) derive the general expression for maximum height hmax and the horizontal range R. b) For what value of θ gives the highest maximum height?Solut