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Vertical motion

Graphical representation of our solution for B=0

Height y(t) as a function of the time t, of an object which is moving in the vertical direction under the influence of the gravitational attraction near the surface of the Earth and, moreover, under the influence of a frictional force proportional to its velocity.		Velocity v(t) as a function of the time t, of an object which is moving in the vertical direction under the influence of the gravitational attraction near the surface of the Earth and, moreover, under the influence of a frictional force proportional to its velocity.

In the above figures we represent graphically the motion of an object which is vertically launched with an initial velocity of 30 m/s while immersed in a viscous liquid with A=-2/s as it is given by our solution for the dynamical equation of objects moving freely and vertically in viscous media under the influence of the gravitational attraction near the surface of the Earth.

In the lefthand-side figure we show how the height varies with time. The figure sets out at t=0 seconds where the height of the ball equals y=0 metres and where its initial upward speed is measured to be equal to 30 m/s. At first the object rises fast in time, but then it slows rapidly down untill, at about 1.0 second after being launched, it reaches for t=0 s its maximum position at about 10 metres height. From thereon the object starts falling towards the Earth with increasing velocity, thereby reducing its height.
Notice that the same object in vacuum would have reached a height of about 46 metres after about 3.1 seconds.
The t-y graph shows an almost straight-line curve for instants of time larger than t=1 s, indicating that the object has reached the terminal velocity given by v_term=-g/A=4.9 m/s.

In the righthand-side figure we show how the velocity varies with time. It sets out with an upward speed of 30 m/s which within one second is reduced to zero. Then it gains downward speed during the first second. After the instant of time t=1 s the speed reaches the terminal value v_term.

A=0