Los Altos - AP Physics C

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AP Physics C - Motion with Air Resistance

A ball of mass Mis thrown vertically upward with an initial speed of v0. It experiences a force of air resistance givenby F = -bv, where b is a positive constant. The positive direction for all vector quantities is upward. Express allalgebraic answers in terms of M, b, v0, and fundamental constants.

a. Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball movesupward?

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 theball 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 theheight 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'sflight, where tf is the time at which the ball returns to the height from which it was thrown.

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AP Physics - Kinematics in Drag, f = -bV

Name

Period Date

A rubber ball of mass m is dropped from a cliff. As the ball falls, it is subject to air drag (a resistiveforce caused by the air). The drag force on the ball has magnitude bv2, where b is a constant dragcoefficient and v is the instantaneous speed of the ball. The drag coefficient b is directly proportional tothe 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.

~» jt,a. On the figure to the right, draw and label all the forces on the ^ j • " O V

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, orremains 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 oftime t, the given quantities, and fundamental constants.

d. Determine the terminal speed vt in terms of the given quantities and fundamental constants.

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Los Altos - AP Physics CName

Period Date

AP Physics - Gravity, Energy & Oscillations

A spherical, non-rotating planet has a radius R and a uniform density p throughout its volume. Suppose anarrow tunnel were drilled through the planet along one of its diameters, as shown in thefigure above, in which a small ball of mass m could move freely under the influence offgravity. Let r be the position of the ball from the center of the planet

/a. Show that the magnitude of the force on the ball at a distance r < R from the It-

center of the planet is given by Fg = - C r , where C = 4/3 (TI Q p m).

The ball is dropped into the tunnel from rest at point P at the planet's surface.

b. Fully describe the subsequent motion of the ball from the time it reaches the center of the planet.

c. Derive and solve a differential equation that could be used to calculate the time it takes the ball tomove from point P to the center of the planet.

d. Determine the time from point P to the center of the planet in term of the initial conditions given.

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