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4.3.1: Elementary Mechanics (Exercises)

• • William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathematics) 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.

Q4.3.1

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.

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.

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.

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 $$t > 0$$.

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.

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.

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.

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

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

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 4.3.8 to find $$y_m$$.

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.

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.

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.

1. Find the time $$T$$ when the object reaches its maximum altitude.
2. Use the result of Exercise 4.3.11 to find the velocity of the object for $$t > T$$.

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.

1. Set up a differential equation for the speed of the object.
2. 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 $$a>a_0$$.

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 $$(0,\infty)$$.

1. 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).
2. Show that if $$\lim_{s\to\infty}f(s)\le mg$$ then $$\lim_{t\to\infty}s(t)=\infty$$.
3. Show that if $$\lim_{s\to\infty}f(s)>mg$$ then $$\lim_{t\to\infty}s(t)=s_T$$ (terminal speed), where $$f(s_T)=mg$$..

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.

1. Set up the initial value problem for the velocity $$v$$ of the mass for $$t>0$$.
2. Use Exercise 4.3.14 (c) to determine the terminal velocity of the object.
3. 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 4.3.3.

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.

1. Set up the initial value problem for the velocity $$v$$ of the mass for $$t>0$$.
2. Use Exercise 4.3.14 (c) to determine the terminal velocity of the object.
3. 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 4.3.3.

Q4.3.2

In Exercises 4.3.17-4.3.20, assume that the force due to gravity is given by Newton’s law of gravitation. Take the upward direction to be positive.

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.

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.

19.

1. Show that (Equation 4.3.27) can be rewritten as $v^2={h-y\over y+R} v^2_e+v_0^2.\nonumber$
2. 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 $y_m={h+R\rho^2\over 1-\rho^2}.\nonumber$
3. By requiring that $$v(y_m)=0$$, use (Equation 4.3.26) 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 $$y \ge h$$.
4. 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 $$y=h$$.

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 $$t>0$$.