Question

At low speeds (especially in liquids rather than gases), the drag force is proportional to the speed rather than it's square, i.e., F = -c_1*r*v , where c_1 is a constant. At time t = 0, a small ball of mass is projected into a liquid so that it initially has a horizontal velocity of in the direction as shown. The initial speed in the vertical direction ( y) is zero. The gravitational acceleration is g. Consider the cartesian coordinate system shown in the figure (+x to the right and +y downwards).

Express the answer of the following questions in terms of some or all of the variables c_1, r , m , g, v_x , v_y , and u (enter C_1 for , v_x for and v_y for ). Enter e^(-z) for exp(-z) (the exponential function of argument -z).

(a) What is component of the acceleration in the direction as a function of the component of the velocity in the direction ? express your answer in terms of v_x , c_1 , r , g , m and u as needed: What is the acceleration in the direction as a function of the component of the velocity in the direction ? express your answer in terms of v_y , c_1 , r , g, m and as needed: Using your result from part (a), find an expression for the horizontal component of the ball's velocity as a function of time? Using your result from part (b), find an expression for the vertical component of the ball's velocity as a function of time t. How long does it take for the vertical speed to reach 99 of its maximum value.What value does the horizontal component of the ball's velocity approach as t becomes infinitely large?What value does the vertical component of the ball's velocity approach as t becomes infinitely large?

Answer 1

a)(-C_1*r*v_x)/m
b)g-(C_1*r*v_y)/m
c)u*e^(-(C_1*r*t)/m)
d)((m*g)/(C_1*r))-((m*g)/(C_1*r))*e^(-(C_1*r*t)/m)
e)4.6*(m/(C_1*r))
f)0
g)(m*g)/(C_1*r)

Answer 2

At low speeds (especially in liquids rather than gases), the drag force is proportional to the speed rather than it's square, i.e., F = -c_1*r*v , where c_1 is a constant. At time t = 0, a small ball of mass is projected into a liquid so that it initially has a horizontal velocity of in the direction as shown. The initial speed in the vertical direction ( y) is zero. The gravitational acceleration is g. Consider the cartesian coordinate system shown in the figure (+x to the right and +y downwards).

Express the answer of the following questions in terms of some or all of the variables c_1, r , m , g, v_x , v_y , and u (enter C_1 for , v_x for and v_y for ). Enter e^(-z) for exp(-z) (the exponential function of argument -z).

(a) What is component of the acceleration in the direction as a function of the component of the velocity in the direction ? express your answer in terms of v_x , c_1 , r , g , m and u as needed: What is the acceleration in the direction as a function of the component of the velocity in the direction ? express your answer in terms of v_y , c_1 , r , g, m and as needed: Using your result from part (a), find an expression for the horizontal component of the ball's velocity as a function of time? Using your result from part (b), find an expression for the vertical component of the ball's velocity as a function of time t. How long does it take for the vertical speed to reach 99 of its maximum value.What value does the horizontal component of the ball's velocity approach as t becomes infinitely large?What value does the vertical component of the ball's velocity approach as t becomes infinitely large?