A certain airplane has a cruising speed of 200 km/hr relative to the ground when there is no wind. The pilot points the airplane at 40^{o} N of W and flies for three hours.

a. How far north and how far west of his original location is he after the three hours?

b. Answer part (a) above if a 50 km/hr wind blows at 30^{o} N of E during the trip.

**Note:**Please make diagrams to support your work, but solve the problem using vector components.

An object is thrown vertically upward in such a way that it has a speed of 19.6 m/s when it reaches one half its maximum altitude. Find:

a. its maximum altitude.

b. its velocity 1 sec after it is thrown.

c. its acceleration when it reaches its maximum altitude.

A rock is thrown vertically upward with an initial speed of 100 ft/s. At the same instant another rock is thrown vertically downward from the top of a 280 ft cliff with an initial speed of 40 ft/s. Neglecting air friction,

a. Express the height above the around for each rock as a function of time.

b. Find the time when the rocks pass each other.

c. Find the height above the ground at which the rocks pass each other.

d. When are the speeds of the rocks the same?

A motorist drives his car 3 km north, then 2 km northeast (45^{o} north of east) and finally 4 km at 37^{o} south of east. Where does he end up relative to his starting point?

**Note:** Please make diagrams to support your work, but solve the problem using vector components.

The position of a certain particle is known as a function of time to be

x = xo + 12t - 2t^{3}

a. What is the velocity and acceleration of the particle at t = 3 s?

b. At what times is the velocity equal to zero?

Two vectors are given as follows: **A** has a magnitude of 15 units at 30^{o} S of E, **B** has a magnitude of 8 units at 10^{o} N of W.

a. Make a rough sketch of **A** + **B**

b. Find the vector **A** + **B** by the component method. Be sure to show your work.

A motorist is speeding down the highway at 35 m/s when he spots a highway patrol officer 100 m behind him traveling at 40 m/s. The motorist immediately begins slowing down at 4m/s^{2} while the officer continues moving at constant speed.

a. How long will it take for the highway patrol officer to catch the motorist?

b. How fast will the motorist be going when the officer passes him?

A gorilla walks 20 m due north and then walks 30 m due west. At the same time his trainer walks 75 m at 65^{o} S of E.

a. Make a careful vector diagram showing the displacements of the gorilla and the trainer.

b. In what direction and how far away does the gorilla look to see his trainer? Use vector components to solve this problem.

A jetliner lands on a runway at 70 m/s, reverses its engines to provide braking, and comes to a halt 29 s later.

a. What is the acceleration of the jet?

b. What is the minimum length of runway that the jet would need to safely come to a stop.

A boy's hand pushes a ball through a vertical distance of 1.45 m as he throws the ball straight upward. Once the ball leaves his hand it travels an additional 22 m (straight up) before returning to earth. Find the acceleration of the ball while it was in the boy's hand.

The human body can withstand a negative acceleration of 250 m/s^{2}. If you are in an automobile traveling at 96 km/hr (60 mi/hr) and have a collision with a solid object, over what minimum distance must you stop to not exceed this acceleration?

An object is observed to start from rest and move with constant acceleration through a distance of 80.0 meters. An observer carefully measures the time of travel. The experiment is repeated with the object having the same acceleration but now traveling a distance of 120 m. By what percentage does the time increase for the second experiment?

Two vectors are given as follows:

**A** has a magnitude of 15 at 30^{o} S of E, and
**B** has a magnitude of 8 at 10^{o} N of W

a. Make a rough sketch **A** + **B**.

b. Find the vector **A** + **B** by the component method. Be sure to show your work.

c. Find **A** x **B**

d. Find **A** "dot" **B**

A car traveling down the highway at a safe speed suddenly starts braking at constant acceleration and comes to a stop in a distance of 60 m. Using the **same braking force** (same acceleration), what would be the stopping distance for the car if the initial speed were increased by 50%?

A car accelerates from rest at a stop sign at 2.00 m/s^{2} for half of the distance to the next stop sign and then decelerates at this same rate for the final half. If the stop signs are 900 m apart,

a. find the time of travel between stop signs

b. find the maximum speed of the car

Bill is driving his motorcycle along a straight road at 25 m/s when he suddenly sees a cow in the road directly in front of him at a distance of 110 m. It takes 0.7 sec for his hand to squeeze the brake lever (the reaction time) and then his cycle slows down at -4 m/s^{2}.

a. How far does he travel in coming to a stop (from the point where he first sees the cow)?

b. What should his reaction time be if he wanted to stop in a distance of exactly 110 m?

A frustrated physics student throws her physics text straight down with a speed of 15 m/s from a 50 m high cliff above Lake Tahoe. Neglecting air friction,

a. How fast is the book traveling by the time it reaches the water?

b. How much time would it take for the book to reach the water?

c. If instead she had thrown the book straight up at 15 m/s, how fast would it be traveling when it reached the water?

A man throws a flower pot upward with a velocity of 4 m/s from a window ledge 8 m above the ground.

a. How much time does it take for the flower pot to hit the ground?

b. How fast is the flower pot traveling just before it hits the ground?

A young woman named Kathy Kool buys a sports car that can accelerate at
6 m/s^{2}. She decides to test the car by racing another speedster, Sneaky Stan. Stan is so sneaky that he manages to leave the starting line with an initial velocity of 2.5 m/s while Kathy leaves the same point from rest. If Stan accelerates at 5 m/s^{2}, find

a. the time it takes Kathy to catch Stan.

b. the distance she travels before she catches him.

c. the velocities of both cars at the instant she catches Stan.

A ball is thrown vertically upward with an initial speed of 19.6 m/s. Sketch a graph for the position, velocity, and acceleration of the ball for the first five seconds of its motion.
A woman with a package in her hand is riding in a hot air balloon which is climbing at a constant speed of 8.0 m/s. Holding the package out over the edge of the gondola, she releases it at the instant that it is 96 m above the ground.

a. How long does it take for the package to hit the ground?

b. How fast is the package going the instant before it hits the ground?

A car with new tires can stop (with constant acceleration) in a distance of 50 m when it is traveling at the speed limit on a certain highway. Assuming the same acceleration, what is the stopping distance if the speed is increased by 25 percent?

Two
blocks are fastened to the ceiling of an elevator with massless ropes
as shown. The elevator accelerates upward at 3 m/s^{2}

a. Make a free body diagram for each of the two masses. Clearly label all forces.

b. Find the tension in each rope.

A 20
ky box is pushed across a floor with a force of 80 N as shown. The coefficient
of kinetic friction between the box and the floor is 0.25. Find the acceleration
of the box.

A 1500 kg car travels around a 300 m radius curve travelling at a speed of
22 m/s. What is the minimum coefficient of friction between the tires and the
road so that the car can stay on the road. Assume that the road is level.

A physics student holds a 1.50 m long string attached to a 2.00 kg rock and whirls the string so that the rock follows a circular path in a **vertical** plane. The speed of the rock is such that the rock completes each revolution in 1.40 s.

a. Make a free body diagram of the rock at the position shown
in the diagram.

b. Find the tension in the string when the rock is at the position shown in the diagram.

A certain gun is fired while pointing vertically upward and the bullet is observed to reach a maximum height of 700 m. Neglecting air friction,

a. find the maximum height of the bullet if, instead, the same gun were fired at an angle of 35^{o} with the horizontal.

b. How long would the bullet stay in the air for this second case (part a)?

A ferris-wheel carries its riders in a vertical circle with a radius of 8.0 m. The ferris-wheel makes one revolution every 9.0 s. Find the apparent weight of a 70 kg person when he is at the lowest point of the circle.

A pitcher throws a ball at an angle of 37^{o} with the horizontal and observes that the ball stays in the air for 2.5 s before hitting the ground. negleching air fricton and the height of the pitcher, find

a. the initial speed of the ball.

b. maximum height reached by the ball.

c. How fast would the pitcher have to run (at constant speed) to catch his own ball?

A 24 kg box is pulled with
a rope along a horizontal floor as shown. The coefficient of sliding friction
is 0.35 and the breaking force of the rope is 520 N. What is the shortest amount
of time that the box can be pulled a distance of 20 m across the floor?

3.0 s after a projectile is fired into the air from the ground, it is observed to have a velocity **v** = (7.6 **i** + 4.8 **j**) m/s where **i** and **j** are the usual unit vectors in the x (horizontal) and y (vertical) directions. Find,

a. the maximum height reached by the projectile.

b. the velocity (direction and magnitude) of the projectile the instant before it hits the ground.

c. the acceleration of the projechle at the top of its path.

A
10 kg block is pushed up a plane inclined at 30^{o} with the horizontal
as shown. The force P = 100 N and is horizontal. The kinetic coefficient of
friction between the two surfaces is 0.5 . Find the acceleration of the block.

A 300 g mass on a horizontal surface is pushed against a spring until the spring is compressed a distance of 30 cm. The mass is then released and is pushed (by the spring) towards a 50 cm radius circular loop. Assuming all surfaces are frictionless, what minimum spring constant is necessary for the mass to make it successfully through the loop?

A 4 kg block is given an initial downward speed of 4 m/s at the top of a 35^{o }incline. The frictional force that opposes the motion is 30 N. Use conservation of energy to find the distance along the incline that the block moves before it stops.

Two objects (m1 = 2.15 kg and m2 = 4.30 kg) sit on a horizontal, frictionless suface and are pushed together with a compressed spring between them. The spring has a spring constant of 154 N/m and the spring is compressed a distance of 2.5 m. The masses are then released.

a. How much energy was stored in the spring?

b. Assuming that all of the stored energy is transferred to the massses and that neither mass is attached to the spring after they are released, find the velocity of each mass.

Three uniform rods are attached perpendicular to each other to form a block letter "C". The top and bottom rods of the "C" are 3 m long and each have a mass of 2 kg. The vertical part of the "C"is 4 m long and has a mass of 5 kg. Find the center of mass of the system.

Water pours from a faucet at the rate of 950 cm^{3}/min. Knowing the density of water to be 1 g/cm^{3}, find the average force exerted by the water on a surface 1.5 m below the faucet. Assume the water comes to rest after hitting the surface.

A 50 kg board, of length 7 m, rests on a table with 3 m of its length hanging over the edge of the table. Is it possible for a 10 kg child to stand on the overhanging end of the board without tipping the board? Explain.

A 160 lb phyiscs student can throw a 1 lb potato with a velocity of 50 ft/s. If he stands on a frictionless frozen lake holding a sack of these 1 lb potatoes (assume the mass of the potatoes is very small comared to the student), how many potatoes per second should he throw to acclerate himself at 2 ft/s^{2}?

**Note:**In the following use energy methods in your solution.

A 3 kg object sitting at the top of the track shown is given an initial velocity of 4 m/s. Assuming that the track is perfectly frictionless between points A and C,

a. Find the velocity of the object at point B.

b. The track continues on horizontally after point C, but now has a coefficient of kinetic friction of 0.4. How far does the object travel after passing point C?

A
2 kg uniform rod of length 1m is constrained to rotate about a horizontal axis
passing through its center. The rod has a 1.5 kg point mass attached to one
end and a 0.5 kg point mass attached to the other end. The 1.5 kg mass is given
a downward push so that it starts moving with an initial velocity of 1.2 m/s.

a. Find the moment of inertia of the system.

b. Find the initial kinetic energy of the system.

c. Through what total angle does the system rotate through before coming to rest?

A 3 kg mass attached to a spring undergoes SHM on a frictionless horizontal
surface. The period of the motion is 4.5 s. If the mass was originally pulled
back 12 cm from the equilibrium position, find the

a. total energy of the system.

b. the maximum velocity of the mass.

c. the position of the mass when one half the total energy is in potential form.

d. Write an equation for the position of the mass as a function of time.

A satellite of mass m is to be placed in a circular orbit so that the satellite is a distance of 2R_{e} above the surface of the earth. (Leave all answers in terms of R_{e}, M_{e}, and G.)

a. What minimum amount of work must be done to lift the satellite into this orbit? Hint: Find the change in energy.

b. What minimum initial speed would the satellite need to leave from its orbit and completely escape from the earth?

A solid disk of mass M and radius R is free to rotate about a fixed axis through
the point O which passes through the center of mass of the disk. A mass m hangs
by a string which is wrapped around the rim of the disk. If the system starts
from rest, find the velocity of the hanging mass after it has fallen through
a vertical height h. Solve the problem by using

a. Newton's 2nd Law

b. Conservation of energy

A sphere of radius 0.4 m is released from rest and rolls down an inclined plane
as shown.

a. Find the velocity of the center of mass of the sphere when it has traveled
a distance of 8 m along the incline. Use conservation of energy in your solution.

b. From your answer in part a determine the angular acceleration experienced
by the sphere.

What would happen to the acceleration of gravity at the surface of the earth
if the mass of the earth increased by 20% and its radius decreased by 20%?

A disk-shaped cloud of hydrogen with mass 10^{30} kg and radius 10^{8}
km rotates about an axis through the center of the disk as shown with a period
of 1000 days. After a long time, the disk contracts into a spherical star of
radius 10^{6} km. What is the new period of rotation?