A flea jumping on level ground leaves the ground with a velo

A flea jumping on level ground leaves the ground with a velocity directed at an angle theta to the ground. At the top of its jump, it is at a height h above the ground. Describe the path of the flea if air resistance is negligible. If the take-off velocity v is fixed, what angle theta should the flea jump to reach maximum horizontal distance At what angle theta should the flea jump to reach a given distance X

Solution

a. the path of the flea is a projectile path with x(t) = V * T * cos(q) horizantal component of velocitu

     with q being angle of projection

     and heigh at time t y(t) = V * T * sin(q) - gT^2 / 2 , form the relation S = ut +at^2 / 2 here velocity and accleration are oppsite in direction

after a time T the flee projectile reach ground and y = 0 ; from the above realation y = 0 => T = 2 V sinq / g

      

b. In time T the horizantal distance covered X = V^2 sin(2q)/ g ; using T at y = 0 in the equation for x(T)

                                       it is clear that sin(2q) is maximum when q =45 ; hence when the flee jump 45 angle it will reach maximum horizantal distance

c )    cos (q) = X / V * T      with T = time of flight = 2Vsin(q)/g