A projectile is fired up an incline (incline angle $\phi$) with an initial speed vi at an angle $\theta$i with respect to the horizontal ($( \theta i > \phi )$). (a) Show that the projectile travels a distance d up the incline, where(b) For what value of $\theta$i is d a maximum, and what is that maximum value? $d = \frac { 2 v _ { i } ^ { 2 } \cos \theta _ { i } \sin \left( \theta _ { i } - \phi \right) } { g \cos ^ { 2 } \phi }$

A 1.0-kg ring with an inner radius of 0.060 m and an outer radius of 0.080 m is sent rolling without slipping up a ramp that makes an angle of $30 ^ { \circ }$ with the horizontal. If the initial speed of the ring is 2.8 m/s, how far along the ramp does it travel before it comes to a stop?

select the best answer from the four alternatives. Which of these best describes the index of refraction for a medium? (a) the speed of light in a vacuum times the speed of light in the medium (b) the speed of light in a vacuum minus the speed of light in the medium (c) the speed of light in a vacuum divided by the speed of light in the medium (d) the speed of light in a vacuum plus the speed of light in the medium

The initial speed of a 2.4-kg box traveling up a plane inclined $37^{\circ}$ to the horizontal is $3.8 \mathrm{~m} / \mathrm{s}$. The coefficient of kinetic friction between the box and the plane is 0.30. How far along the incline does the box travel before coming to a stop?

As an object moves at a speed close to the speed of light, length measured in the direction of the speed contracts.

Nate throws a ball straight up to Kayla, who is standing on a balcony $3.8 \mathrm{~m}$ above Nate. When she catches it, the ball is still moving upward at a speed of $2.8 \mathrm{~m} / \mathrm{s}$. With what initial speed did Nate throw the ball?
A. $7.0 \mathrm{~m} / \mathrm{s}$
B. $12.3 \mathrm{~m} / \mathrm{s}$
C. $9.1 \mathrm{~m} / \mathrm{s}$
D. $10.6 \mathrm{~m} / \mathrm{s}$

A 60-kg housepainter stands on a 15-kg aluminum platform. The platform is attached to a rope that passes through an overhead pulley, which allows the painter to raise herself and the platform (Figure 4-57).

$(b)$ When her speed reaches $1.0 \mathrm{~m} / \mathrm{s}$, she pulls in such a way that she and the platform go up at a constant speed. What force is she exerting on the rope now? (Ignore the mass of the rope.)

A $1.5\ \mathrm{kg}$ object moves up the y-axis at a constant speed . When it reaches the origin, the forces $F_1 = 5.0\ \mathrm{N}$ at $37^\circ$ above the +x -axis, $F_2 = 2.5\ \mathrm{N}$ in the +x -direction , $F_3 = 3.5\ \mathrm{N}$ at $45^\circ$ below the - x-axis, and $F_4 = 1.5\ \mathrm{N}$ in the -y -direction are applied to it. (a) Will the object continue to move along the y-axis? (b) If not, what simultaneously applied force will keep it moving along the y-axis at a constant speed?

When the 5 -kg box reaches point A it has a speed $v_{A}=10$ m/s. Determine how high the box reaches up the surface before it comes to a stop. Also, what is the resultant normal force on the surface at this point and the acceleration? Neglect friction and the size of the box.

A carpenter on the roof of a building accidentally drops her hammer. As the hammer falls it passes two windows of equal height, as shown in Figure 2-32. Choose the best explanation from among the following:

I. The greater speed at window 2 results in a greater increase in speed.

II. Constant acceleration means the hammer speeds up the same amount for each window.

III. The hammer spends more time dropping past window 1.

A double speed trap is set up on a freeway. One police cruiser is hidden behind a billboard, and another is some distance away under a bridge. As a sedan passes by the first cruiser, its speed is measured as $105.9 \mathrm{mph}$. Since the driver has a radar detector, he is alerted to the fact that his speed has been measured, and he tries to slow his car down gradually without stepping on the brakes and alerting the police that he knew he was going too fast. Just taking the foot off the gas leads to a constant deceleration. Exactly $7.05 \mathrm{~s}$ later, the sedan passes the second police cruiser. Now its speed is measured as only $67.1 \mathrm{mph}$, just below the local freeway speed limit. a) What is the value of the deceleration?

Motive power for the experimental 10-Mg bus comes from the energy stored in a rotating flywheel which it carries. The flywheel has a mass of 1500 kg and a radius of gyration of 500 mm and is brought up to a maximum speed of 4000 rev/min. If the bus starts from rest and acquires a speed of 72 km/h at the top of a hill 20 m above the starting position, compute the reduced speed N of the flywheel. Assume that 10 percent of the energy taken from the flywheel is lost. Neglect the rotational energy of the wheels of the bus. The 10-Mg mass includes the flywheel.

The elevator in Yokohama's 70-story Landmark Tower operates at a peak speed of $45 \mathrm{~km} / \mathrm{hr}$. To reach such a speed without inducing discomfort in passengers, the elevator accelerates for longer periods, rather than more precipitously. Going up, it reaches full speed only at the 27 th floor; it begins decelerating 15 floors later. The result is a peak acceleration similar to that of other skyscraper elevators-a bit less than a tenth of the force of gravity. Admirable ingenuity has gone into making this safe and comfortable. Special ceramic brakes had to be developed; iron ones would melt. Computer-controlled systems damp out vibrations. The lift has been streamlined to reduce the wind noise as it speeds up and down . One proposed control system for the elevator vertical position is shown in Figure . Determine the range of $K$ for a stable system.

An object is projected with initial speed $v_{0}$ from the edge of the roof of a building that has height H. The initial velocity of the object makes an angle $\alpha_{0}$ with the horizontal. Neglect air resistance. If $\alpha_{0}$ is 90$^{\circ}$, so that the object is thrown straight up (but misses the roof on the way down), what is the speed v of the object just before it strikes the ground?