# Calculus Based Physics

A ball of mass M is thrown vertically upward with an initial speed of v0. 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, v0, and fundamental constants.

Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball moves upward?

Write, but do NOT solve, a differential equation for the instantaneous speed vof the ball in terms of time tas the ball moves upward.

Determine the terminal speed of the ball as it moves downward.

Does it take longer for the ball to rise to its maximum height or to fall from its maximal height back to the height from which it was thrown? Justify your answer.

Sketch a graph of velocity versus time for the upward and downward parts of the ball’s flight, where $t_f$ is the time at which the ball returns to the height from which it was thrown.

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## Related Problems

An Object with mass m and initial velocity v is brought to rest by a constant force F acting for a time t and through a distance d. Possible expressions for the magnitude of the force F are: i. $\frac{mv^2}{2d}$ ii. $\frac{2md}{t^2}$ iii. $\frac{mv}{t}$

ii only

iii only

i and ii only

ii and iii only

i, ii, and iii

A toy car of mass 6 kg moving in a straight path, experiences a net force given by the function F = -3t. At time t=0, the car has a velocity of 4 m/s in the positive direction and is located +8 m from the origin. The car will come instantaneously to rest at time t equal to

2/3 s

sqrt( 4/3 ) s

sqrt( 8/3 ) s

sqrt( 8) s

4 s

A small block of mass MB= 0.50 kg is placed on a long slab of mass MS= 3.0 kg as shown below. Initially, the slab is at rest and the block has a speed v0of 4.0 m/s to the right. The coefficient of kinetic friction between the block and slab is 0.20, and there is no friction between the slab and the horizontal surface on which it moves.

A box is being pulled with force Fa= 40 N at an angle θ= 30 degrees. There is a frictional force opposing the sliding of the box with force Ff= 10 N. Draw a free body diagram to help answer the following questions. (use gravity at 10m/s2 to simplify the numbers)