4.3E: Elementary Mechanics (Exercises)
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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 lb-s2/ft2. 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)=y0. Use the results of Example 4.3.1 to show that
y(t)=y0+mk(v0−v−gt).
9. An object with mass m is launched vertically upward with initial velocity v0 from Earth’s surface (y0=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 ym. Then use the result of Exercise 4.3.8 to find ym.
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 ft/s. Find v for t>0, and find its terminal velocity.
11. An object with mass m is given an initial velocity v0≤0 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 v0 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 4.3.11 to find the velocity of the object for t>T.
13. An object with mass m is given an initial velocity v0≤0 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 a0 such that limt→∞s(t)=∞ if a≤a0 and limt→∞s(t) exists (finite) if a>a0. (We say that a0 is the bifurcation value of a.) Try to find a0 and limt→∞s(t) in the case where a>a0.
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,∞).
- 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)≥0 are defined for all t>0 (which makes good sense on physical grounds).
- Show that if lims→∞f(s)≤mg then limt→∞s(t)=∞.
- Show that if lims→∞f(s)>mg then limt→∞s(t)=sT (terminal speed), where f(sT)=mg..
15. A 100-g mass with initial velocity v0≤0 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 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 v0=0, −2, −4, …, −12. Present your results in graphical form similar to Figure 4.3.3.
16. A 64-lb object with initial velocity v0≤0 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 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 v0=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/s2. Find the escape velocity in miles/s.
19.
- Show that (Equation 4.3.27) can be rewritten as v2=h−yy+Rv2e+v20.
- Show that if v0=ρve with 0≤ρ<1, then the maximum altitude ym attained by the space vehicle is ym=h+Rρ21−ρ2.
- By requiring that v(ym)=0, use (Equation 4.3.26) to deduce that if v0<ve then |v|=ve[(1−ρ2)(ym−y)y+R]1/2, where ym and ρ are as defined in (b) and y≥h.
- Deduce from (c) that if v<ve, the vehicle takes equal times to climb from y=h to y=ym and to fall back from y=ym to y=h.
20. In the situation considered in the discussion of escape velocity, show that limt→∞y(t)=∞ if v(t)>0 for all t>0.