Question

1. A ball of mass m is thrown with speed vo at an angle 0 with respect to the horizontal. Let the drag force from the air take the form f Newton's second law, the r- and y-components of the acceleration of the ball are given by = mav where a is a constant and v is the speed of the ball. By du (t) = -av,(t), dt de,(t) ay(t)= -g-au,(t), dt where ug (t) and v,(t) are the x- and y-components of the velocity, and g is the magnitude of accel- eration due to gravity. Assume that the magnitude of the initial drag force equals to the weight of the ball, determine the value of 0 so that the value of r is as large as possible when y achieves its maximum value. [30 pts]

Answer 1

Ans dv ) - dt Vx te both sidees LnVx -dt tc dt -i Cie VX

dvy 1 -dt both sides ate Inlj+avy) t tc tt tVyC2e +

Vy= C2et Now as given at intial daye forite Menithde puat to aof 1he batt thus md V Vo Coso E Vo Sine At t-o thus C e CLE VoCose VD Sin= Vo Sine t 7ltSine ht -9t Vx Valose e I+ Sine) e

Date Page -vot - Vo Vo (t+Sine) e 74/B - 1 VlltSine)e for maxirmum Vy=o Itsine) e ot I+ Sine) -Not Vo COse Thtot dx Vocoso e - 9/vt Vo Cose e lt) +C

Vo Coso x lt)-0 = at t-o -/vot 2 at 4 maximum Condition x lt)- 2 Vo Coso 7 1t Sine Sino 2 = Sin20 1+Sine) tSine)

with & for X-Maximum dx de SinRO 1 CosE 2 Cos20 6) VD l+Suney2 Sine) /it Sine) Cob26 Sin 28 Cose

Page RANKA Sino cose l+Sine) Coo COBRD tSine Cos2e Sine loso 1 Sime Cos20- Cos O Cos20 t RCOse-1 Coso Cos20 t Sine 2 Cos20 t Sin0 cob Sine (-Sin'e) Coszo t

- 2 Sin'o-Sin30 3 X XSine 0 + 2x x 0.61803 0= 38.17