Question

A 50 kg object is shot from a cannon straight up with an initial velocity of 10m/s off a bridge that is 100 meters above the ground. If air resistance is given by 5v' determine the velocity of the mass at any time t.
Please solve using differential equations methods, thanks.

Answer 1

we know that F = F_G + F_A where F = mv' F_G = mg & F_A = a^2 where m is mass of object = 50 kg g is force = 9.8 m/s^2 a is (damping constant) air resistance = 5v^2 up Down F_G = mg F_A = 5v^2 F_A = -5v^2 mv' = mg + 5v^2 mv' = mg - 5v^2 v' = g + 5/m v^2 v' = g - 5/m v^2 v' = 9.8 + 1/10 v^2 v' = 9.8 - 1/10 v^2 v(0) = -10 v(10) = 0 upward motion: consider v' = 9.8 + 1/10 v^2 rightarrow (1) v(0) = -10 dv/dt = 9.8 + 1/10 v^2 integral 1/9.8 + 1/10 v^2 dv = integral 1 dt 10 integral 1/9.8 + v^2 dv = integral 1 dt 10 integral 1/(squareroot 98)^2 + v^2 dv = integral 1 dt 10(1 squareroot 98 tan^- 1(v/squareroot 98)) = t + c 10/squareroot 98 tan^- 1 (v(t)/squareroot 98)) = t + c rightarrow (2) is solution of (1) G.T v(0) = -10 10/squareroot 98 tan^- 1(V(0)/squareroot 98) = 0 + c

Answer 2

fkng hand written