Download Homework - Ryan, Susan

AP Physics C - Force: Drag Problems
Notice that in this problem there is no applied force, and the initial conditions are different so the object is slowing down
rather than speeding up:
1. A generic body of mass m is moving along the x-axis with velocity v, and is then slowed by a force in the opposite
direction to the motion, F = -kv, where k is a constant. At time t = 0, the object has velocity vo at position x = 0.
a. What is the initial acceleration (magnitude and direction) produced by the resistance force?
b. Derive an equation for the object's velocity as a function of time t, and sketch this function on the axes
below. Let a velocity directed to the right be considered positive. Hint: This problem is easier since you
can directly integrate the expression without substituting variables.
c. Derive an equation for the distance the object travels as a function of time t and sketch this function on the axes
below. Hint: Now you need to show velocity as the derivative of the position, and integrate the expression for the
velocity to obtain the expression for position.
d. Determine the distance the object travels from t = 0 to t = .
2. An object of mass m near the Earth’s surface falls from rest in the Earth's gravitational field. Acting on the object is a
resistive force of magnitude kmv, where k is a constant and v is the speed of the object.
a.
On the diagram below, draw and identify all of the forces acting on the object as it falls.
b. Write the differential equation that represents Newton's second law for this situation.
c. Determine the terminal speed vT of the object.
d. Integrate the differential equation once to obtain an expression for the speed v as a function of time t. Use the
condition that v = 0 when t = 0.
e. On the axes provided below, draw a graph of the speed v as a function of time t.
3. A rubber ball of mass m is dropped from a cliff. As the ball falls, it is subject to air drag (a resistive force caused by the air). The
drag force on the ball has magnitude bv2, where b is a constant drag coefficient and v is the instantaneous speed of the ball. The drag
coefficient b is directly proportional to the cross-sectional area of the ball and the density of the air and does not depend on the mass
of the ball. As the ball falls, its speed approaches a constant value called the terminal speed.
a. On the figure below, draw and label all the forces on the ball at some instant before it reaches terminal speed.
b.
State whether the magnitude of the acceleration of the ball of mass m increases, decreases, or remains the same as the ball
approaches terminal speed. Explain.
c.
Write, but do NOT solve, a differential equation for the instantaneous speed v of the ball in terms of time t, the given quantities,
and fundamental constants.
d.
Determine the terminal speed vt in terms of the given quantities and fundamental constants.
4. A ball of mass M is thrown vertically upward with an initial speed of vo. It experiences a force of air resistance given by F = -kv,
where k is a positive constant. The positive direction for all vector quantities is upward. Express all algebraic answers in terms of M,
k, vo, and fundamental constants.
a. Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball moves upward?
increases
decreases
remains the same
Justify your answer.
b. Write, but do NOT solve, a differential equation for the instantaneous speed v of the ball in terms of time t as the ball moves
upward.
c.
Determine the terminal speed of the ball as it moves downward.
d. Does it take longer for the ball to rise to its maximum height or to fall from its maximum height back to the height from which it
was thrown?
longer to rise
longer to fall
Justify your answer.
e.
On the axes below, sketch a graph of velocity versus time for the upward and downward parts of the ball's flight, where tf is the
time at which the ball returns to the height from which it was thrown.
5. A car of mass m, initially at rest at time t = 0, is driven to the right, as shown above, along a straight, horizontal road with
the engine causing a constant force Fo to be applied. While moving, the car encounters a resistance force equal to -kv, where v
is the velocity of the car and k is a positive constant.
a. The dot below represents the center of mass of the car. On this figure, draw and label vectors to represent all the forces
acting on the car as it moves with a velocity v to the right.
b.
Determine the horizontal acceleration of the car in terms of k, v, Fo, and m.
c.
Derive the equation expressing the velocity of the car as a function of time t in terms of k, Fo, and m.
d.
On the axes below, sketch a graph of the car’s velocity v as a function of time t. Label important values on the vertical
axis.
e.
On the axes below, sketch a graph of the car’s acceleration a as a function of time t. Label important values on the vertical
axis.
6.
A skier of mass M is skiing down a frictionless hill that makes an angle  with the horizontal, as shown in the diagram. The skier starts from rest at
time t = 0 and is subject to a velocity-dependent drag force due to air resistance of the form F = –bv, where v is the velocity of the skier and b
is a positive constant. Express all algebraic answers in terms of M, b, , and fundamental constants.
(a)
On the dot below that represents the skier, draw a free-body diagram indicating and labeling all of the forces that act on the skier while the
skier descends the hill.
(b)
Write a differential equation that can be used to solve for the velocity of the skier as a function of time.
(c)
Determine an expression for the terminal velocity vT of the skier.
(d)
On the axes below, sketch a graph of the acceleration a of the skier as a function of time t, and indicate the initial value of a. Take
downhill as positive.
a
O
t