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# 4.3E: Elementary Mechanics (Exercises)

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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Except where directed otherwise, assume that the magnitude of the gravitational force on an object with mass $$m$$ is constant and equal to $$mg$$. In exercises involving vertical motion take the upward direction to be positive.

[exer:4.3.1] A firefighter who weighs 192 lb slides down an infinitely long fire pole that exerts a frictional resistive force with magnitude proportional to his speed, with $$k=2.5$$ lb-s/ft. Assuming that he starts from rest, find his velocity as a function of time and find his terminal velocity.

[exer:4.3.2] A firefighter who weighs 192 lb slides down an infinitely long fire pole that exerts a frictional resistive force with magnitude proportional to her speed, with constant of proportionality $$k$$. Find $$k$$, given that her terminal velocity is -16 ft/s, and then find her velocity $$v$$ as a function of $$t$$. Assume that she starts from rest.

[exer:4.3.3] A boat weighs 64,000 lb. Its propellor produces a constant thrust of 50,000 lb and the water exerts a resistive force with magnitude proportional to the speed, with $$k=2000$$ lb-s/ft. Assuming that the boat starts from rest, find its velocity as a function of time, and find its terminal velocity.

[exer:4.3.4] A constant horizontal force of 10 N pushes a 20 kg-mass through a medium that resists its motion with.5 N for every m/s of speed. The initial velocity of the mass is 7 m/s in the direction opposite to the direction of the applied force. Find the velocity of the mass for $$t > 0$$.

[exer:4.3.5] A stone weighing 1/2 lb is thrown upward from an initial height of 5 ft with an initial speed of 32 ft/s. Air resistance is proportional to speed, with $$k=1/128$$ lb-s/ft. Find the maximum height attained by the stone.

[exer:4.3.6] A 3200-lb car is moving at 64 ft/s down a 30-degree grade when it runs out of fuel. Find its velocity after that if friction exerts a resistive force with magnitude proportional to the square of the speed, with $$k=1\ \mbox{lb-s}^2/{\mbox ft}^2$$. Also find its terminal velocity.

[exer:4.3.7] A 96 lb weight is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the speed. Find its velocity as a function of time if its terminal velocity is -128 ft/s.

[exer:4.3.8] An object with mass $$m$$ moves vertically through a medium that exerts a resistive force with magnitude proportional to the speed. Let $$y=y(t)$$ be the altitude of the object at time $$t$$, with $$y(0)=y_0$$. Use the results of Example 4.3.1} to show that

$y(t)=y_0+{m\over k}(v_0-v-gt).\nonumber$

[exer:4.3.9] An object with mass $$m$$ is launched vertically upward with initial velocity $$v_0$$ from Earth’s surface ($$y_0=0$$) in a medium that exerts a resistive force with magnitude proportional to the speed. Find the time $$T$$ when the object attains its maximum altitude $$y_m$$. Then use the result of Exercise [exer:4.3.8} to find $$y_m$$.

[exer:4.3.10] An object weighing 256 lb is dropped from rest in a medium that exerts a resistive force with magnitude proportional to the square of the speed. The magnitude of the resisting force is 1 lb when $$|v|=4\ \mbox{ft/s}$$. Find $$v$$ for $$t > 0$$, and find its terminal velocity.

[exer:4.3.11] An object with mass $$m$$ is given an initial velocity $$v_0\le0$$ in a medium that exerts a resistive force with magnitude proportional to the square of the speed. Find the velocity of the object for $$t > 0$$, and find its terminal velocity.

[exer:4.3.12] An object with mass $$m$$ is launched vertically upward with initial velocity $$v_0$$ in a medium that exerts a resistive force with magnitude proportional to the square of the speed.

Find the time $$T$$ when the object reaches its maximum altitude.

Use the result of Exercise [exer:4.3.11} to find the velocity of the object for $$t > T$$.

[exer:4.3.13] An object with mass $$m$$ is given an initial velocity $$v_0\le0$$ in a medium that exerts a resistive force of the form $$a|v|/(1+|v|)$$, where $$a$$ is positive constant.

Set up a differential equation for the speed of the object.

Use your favorite numerical method to solve the equation you found in (a), to convince yourself that there’s a unique number $$a_0$$ such that $$\lim_{t\to\infty}s(t)=\infty$$ if $$a\le a_0$$ and $$\lim_{t\to\infty}s(t)$$ exists (finite) if $$a>a_0$$. (We say that $$a_0$$ is the bifurcation value of $$a$$.) Try to find $$a_0$$ and $$\lim_{t\to\infty}s(t)$$ in the case where $$a>a_0$$.

[exer:4.3.14] An object of mass $$m$$ falls in a medium that exerts a resistive force $$f=f(s)$$, where $$s=|v|$$ is the speed of the object. Assume that $$f(0)=0$$ and $$f$$ is strictly increasing and differentiable on $$(0,\infty)$$.

Write a differential equation for the speed $$s=s(t)$$ of the object. Take it as given that all solutions of this equation with $$s(0)\ge0$$ are defined for all $$t>0$$ (which makes good sense on physical grounds).

Show that if $$\lim_{s\to\infty}f(s)\le mg$$ then $$\lim_{t\to\infty}s(t)=\infty$$.

Show that if $$\lim_{s\to\infty}f(s)>mg$$ then $$\lim_{t\to\infty}s(t)=s_T$$ (terminal speed), where $$f(s_T)=mg$$..

[exer:4.3.15] A 100-g mass with initial velocity $$v_0\le0$$ falls in a medium that exerts a resistive force proportional to the fourth power of the speed. The resistance is $$.1$$ N if the speed is 3 m/s.

Set up the initial value problem for the velocity $$v$$ of the mass for $$t>0$$.

Use Exercise [exer:4.3.14} (c) to determine the terminal velocity of the object.

To confirm your answer to (b), use one of the numerical methods studied in Chapter 3 to compute approximate solutions on $$[0,1]$$ (seconds) of the initial value problem of (a)

, with initial values $$v_0=0$$, $$-2$$, $$-4$$, …, $$-12$$. Present your results in graphical form similar to Figure [figure:4.3.3}.

[exer:4.3.16] A 64-lb object with initial velocity $$v_0\le0$$ falls through a dense fluid that exerts a resistive force proportional to the square root of the speed. The resistance is $$64$$ lb if the speed is 16 ft/s.

Set up the initial value problem for the velocity $$v$$ of the mass for $$t>0$$.

Use Exercise [exer:4.3.14} (c) to determine the terminal velocity of the object.

To confirm your answer to (b), use one of the numerical methods studied in Chapter 3 to compute approximate solutions on $$[0,4]$$ (seconds) of the initial value problem of (a), with initial values $$v_0=0$$, $$-5$$, $$-10$$, …, $$-30$$. Present your results in graphical form similar to Figure [figure:4.3.3}.

[exer:4.3.17] A space probe is to be launched from a space station 200 miles above Earth. Determine its escape velocity in miles/s. Take Earth’s radius to be 3960 miles.

[exer:4.3.18] A space vehicle is to be launched from the moon, which has a radius of about 1080 miles. The acceleration due to gravity at the surface of the moon is about $$5.31$$ ft/s$$^2$$. Find the escape velocity in miles/s.

[exer:4.3.19]

Show that Eqn. (Equation \ref{eq:4.3.23}) can be rewritten as

$v^2={h-y\over y+R} v^2_e+v_0^2.\nonumber$

Show that if $$v_0=\rho v_e$$ with $$0\le \rho < 1$$, then the maximum altitude $$y_m$$ attained by the space vehicle is

$y_m={h+R\rho^2\over 1-\rho^2}.\nonumber$

By requiring that $$v(y_m)=0$$, use Eqn. (Equation \ref{eq:4.3.22}) to deduce that if $$v_0 < v_e$$ then

$|v|=v_e\left[{(1-\rho^2)(y_m-y)\over y+R}\right]^{1/2},\nonumber$

where $$y_m$$ and $$\rho$$ are as defined in (b) and $$y \ge h$$.

Deduce from (c) that if $$v < v_e$$, the vehicle takes equal times to climb from $$y=h$$ to $$y=y_m$$ and to fall back from $$y=y_m$$ to $$y=h$$.

[exer:4.3.20] In the situation considered in the discussion of escape velocity, show that $$\lim_{t\to\infty}y(t)=\infty$$ if $$v(t)>0$$ for all $$t>0$$.