4.3E: Elementary Mechanics (Exercises)

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 $t>0$.