A projectile of mass m is fired straight up The air drag for

A projectile of mass m is fired straight up. The air drag force on the projectile is kV^2, where V is the speed and k is a positive constant. The drag force is opposite to the motion. The initial velocity of the projectile is V_0. The projectile is fired at time t = 0. Show that the upward velocity becomes zero at a time t_m = squareroot m/gk arctan (squareroot k/gm V_0), where g is the acceleration of gravity. Support m = 3 kg and V_0 = 60 m/s. A measurement shows that t_m = 2.3 s. Use this experimental result to estimate the value of k. Do this by using MATLAB to plot the theoretical expression for t_m as a function of k. Find the maximum height reached by the object in part (b).

Solution

When a projectile makes 600 it makes an angle 300 with the horizontal.the velocity vector of the partical initially is v0=ucos30i+usin30j, after a time t the velocity vector is v=ucos30i+(usin30-gt)j the two vectors are perpendicular so their dot product is zero. we get that ucos30xucos30+usin30xusin30=gtsin30u=>u2=gtsin30=>t=u/gsin30