# Maths

A ball of mass m is thrown vertically upwards with initial speed u, and travels upwards under the influence of gravity and air resistance. Use the quadratic model of air resistance with the ball modelled as a sphere of effective diameter D.

Q1) Apply Newton’s second law to obtain the equationdv/dt = −k(v^2 + g/k)where v is the speed of the ball, t is the time elapsed since the ball was released, g is the magnitude of the acceleration due to gravity, andk = c_2D_2/m is a constant.

Q2)Solve the differential equation and apply the initial condition to find thetime t in terms of the speed v and the constants given above.

Q3) Use your equation to show that the time taken to reach the maximum height attained by the ball, tmax, is given by tmax = 1/√(gk) arctan (√(k/g)u)

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1. I believe your post has not been addressed by a math tutor because you have dumped all this work here, but haven't included any of your own thoughts: how should you solve these? if you get stuck, where? etc. Keep in mind the math tutors won't do all your thinking and work for you.

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