Class Discussion Activity 9

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Exercises

For exercises 1 and 2 use the following information:

A $1$-kg rock is dropped from the top of a large vertical cliff and during free-fall the air resists the motion of the rock proportional to its speed, at a rate of $4$ N of force per m/s of speed.

(1)

Determine the IVP that models the vertical velocity $v(t)$ of the rock at $t\geq 0$ seconds after it is dropped. Use $g=9.8$ m/s$^2$ for the magnitude of the acceleration due to gravity.

(a): $v^{\,\prime }=9.8-4v$, $v(0)=0$
(b): $v^{\,\prime }=9.8+4v$, $v(0)=0$
(c): $v^{\,\prime }=-9.8+4v$, $v(0)=0$
(d): $v^{\,\prime }=-9.8-4v$, $v(0)=0$
(e): $v^{\,\prime }=9.8v+4$, $v(0)=-1$
(f): $v^{\,\prime }=-9.8v-4$, $v(0)=-1$

(2)

Given the rock hits the ground at a speed of $2.42$ meters per second determine the height of the cliff. Round to the nearest meter.

(3)

A $95$ kg skydiver falls through air that resists motion at a rate of $18$ N of force per m/s of speed. Find the terminal velocity in meters per second. Use $g=9.8$ m/s$^2$ for the magnitude of the acceleration due to gravity. Round to the nearest tenth of a meter per second.

(4)

A $10$ kg object falls attached to vertical rails with a friction-breaking device that is designed to exert a resistive force proportional to the fourth power of the speed of the object. The resistance is $96$ N if the speed is $2$ meter per second. Find the terminal velocity in meters per second to the nearest hundredth of a meter per second. Use $g=9.8$ m/s$^2$ for the magnitude of the acceleration due to gravity.

(5)

A $1$ kg stone is sling-shot straight up into the air at $40$ meter per second from an initial height of $1$ meter. Assuming the air resists the stone’s motion at a rate of $1/25$ N of force per m/s of speed, determine the height in meters reached by the object. Use $g=9.8$ m/s$^2$ for the magnitude of the acceleration due to gravity. Round to the nearest tenth of a meter.