4.3E: Elementary Mechanics (Exercises)
- Page ID
- 103493
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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.
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 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.
9. 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.
In Exercises 10 and 11, assume that the force due to gravity is given by Newton’s law of gravitation. Take the upward direction to be positive.
10. A space probe is to be launched from Earth. Determine its escape velocity in miles/s. Take Earth’s radius to be \(3960\) miles.
11. 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.