AP Free Response Question
2008 C1
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A skier of mass
M
is skiing down a frictionless hill that makes an angle
θ
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, θ, 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 v
_{T}
of the skier.
(d) Solve the differential equation in part (b) to determine the velocity of the skier as a function of time, showing all your steps.
(e) 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.
Topic Formulas
Description
Published Formula
angular displacement
angular momentum
angular velocity
center of mass
centripetal acceleration
friction
gravitational force (vector)
gravitational potential energy
gravitational potential energy
Hooke's Law
impulse
kinetic energy
linear momentum
linear velocity and angular velocity
moment of inertia
net torque
Newton's 2nd Law
Newton's Law of Universal Gravitation
period and frequency
period of a simple pendulum
period of a spring
potential elastic energy
potential energy
power (dot product)
rate of change of momentum
rate of change of work
rotational kinetic energy
torque
uniform acceleration - displacement and instantaneous velocity
uniform acceleration - instantaneous position
uniform acceleration - instantaneous velocity
work (dot product)
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Advanced Gravitational Forces
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Air Resistance
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Air Resistance: Terminal Velocity
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Average Velocity - A Calculus Approach
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Inclined Planes
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Inertial vs Gravitational Mass
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Non-constant Resistance Forces
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Properties of Friction
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Springs and Blocks
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Springs: Hooke's Law
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Universal Gravitation and Weight
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Work and Energy
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Equilibrium on an Inclined Plane
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Falling and Air Resistance
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Force and Acceleration
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Force and Weight
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Force Vectors and the Parallelogram Rule
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Freebody Diagrams
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Gravitational Interactions
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Incline Places: Force Vector Resultants
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Incline Planes - Force Vector Components
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Inertia
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Mobiles: Rotational Equilibrium
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Net Force
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Newton's Law of Motion: Friction
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Static Equilibrium
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Tensions and Equilibrium
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Acceleration
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Air Resistance #1
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Apex #2
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Falling Rock
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Falling Spheres
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Friction
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Frictionless Pulley
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Gravitation #1
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Head-on Collisions #1
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Sailboats #1
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Sailboats #2
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Scale Reading
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Settling
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Skidding Distances
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Spiral Tube
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Tensile Strength
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Terminal Velocity
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Tug of War #1
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Tug of War #2
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Two-block Systems
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Advanced Properties of Freely Falling Bodies #1
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Advanced Properties of Freely Falling Bodies #2
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Combining Kinematics and Dynamics
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Distinguishing 2nd and 3rd Law Forces
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Force vs Displacement Graphs
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CB-ETS
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