A body of mass 5 Kg is projected vertically upward with an initial velocity 30 m

Question

A body of mass 5 Kg is projected vertically upward with an initial velocity 30 meters per second. The gravitational constant is g = 9.8m/s^2. The air resistance is equal to k|v| where K is a constant. Find a formula for the velocity at any time (in terms of K): v(t) = Find the limit of this velocity for a fixed time t_0 as the air resistance coefficient k goes to 0. (Write t_0 as to in your answer.) v(t_0) = How does this compare with the solution to the equation for velocity when there is no air resistance? This illustrates an important fact, related to the fundamental theorem of ODE and called 'continuous dependence' on parameters and initial conditions. What this means is that, for a fixed time, changing the initial conditions slightly, or changing the parameters slightly, only slightly changes the value at time t. The fact that the terminal time t under consideration is a fixed, finite number is important. If you consider 'infinite't. or the 'final' result you may get very different answers. Consider for example a solution to y' = y. whose Initial condition is essentially zero, but which might vary a bit positive or negative If the initial condition is positive the "final" result is plus infinity, but if the initial condition is negative the final condition is negative infinity.

Explanation / Answer

to get the velocity of the body which is projected vertically upwards we use the equation of motion

v = u + a t

where v is the final velocity

u is niotial velocity

a is acceleration

t is time taken

in our caae as the ball is projected vertically upwards the final velovity v = 0 and a = -g

where g is acceleration due to gravity

v(t) = u + (- g) t

now if we take the air resistance as also a force which acts on the ball vertically downwards when the ball is going up the equation of motion changes to

v(t) = u + (- g) (k) t

the limit of the velocity will be depending upon the initial velocity u given when ball projected vertically up

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