An object is fired at an angle above horizontal and is subje

An object is fired at an angle above horizontal and is subject to the forces of gravity and air resistance. Gravity only affects the vertical motion of the object but air resistance affects both horizontal and vertical motion. If the drag force is linear, then the differential equations: mv\'_x = -bv_x mv\'_y = mg - bv_y describe the object\'s speed. (a) Classify the differential equations. Are they ordinary or partial differential equations? What order are they? Are they linear or non-linear? (b) The equations could be rewritten as: m_x\" = -bx\' my\" = mg - by\' What order are the equations now? (c) Show that v_x = Ae^-b/m is a solution to mv\'_x = -bv_x. (d) What is the physical meaning of v\'_y = 0? Find v_y when v\'_y = 0. (e) Let v_t be your answer to part (d). Show that v_y = v_t + Be^-b/m is a solution to mv\'_y = mg - b_vy (f) Suppose the object was fired with an initial speed of 200 meters per second at an angle of 30degree above horizontal. Find the values of the constants A and B from parts (c) and (e). (Your answers may depend on m and b.

Solution

(a).These are the ordinary differential equation as ,these depend upon single varibale x that depend upon t.

as v^\' =single differentiation of v, hence order of these equations is one.

These equations are linear equations because, equations donot conatin any varible as a function of \"t\" in numerator.

(b).here x^\'\' = double differentiation of x,hence order of thes equation is two.

(d)dv/dt=0 means v=constant.