An object is released from rest at the top of a ramp. If the ramp is frictionless, does the object’s speed at the bottom of the ramp depend on the shape of the ramp or just on its height? Explain. What if the ramp is not frictionless?

Is it possible for a frictional force to increase the mechanical energy of a system? If so, give examples.

A projectile has the same initial kinetic energy no matter what the angle of projection. Why doesn’t it rise to the same maximum height in each case?

In splitting logs with a hammer and wedge, is a heavy hammer more effective than a lighter hammer? Why?

The ramp in as we discussed earlier is frictionless. If the blocks are released from rest, which way does the 10 kg block slide, and what is the magnitude of its acceleration?

A 1.9-kg block slides down a frictionless ramp. The top of the ramp is 1.5 m above the ground; the bottom of the ramp is 0.25 m above the ground. The block leaves the ramp moving horizontally, and lands a horizontal distance d away. Find the distance d.

A 10.0-g marble slides to the left at a speed of 0.400 m/s on the frictionless, horizontal surface of an icy New York sidewalk and has a head-on, elastic collision with a larger 30.0-g marble sliding to the right at a speed of 0.200 m/s. Find the velocity of each marble (magnitude and direction) after the collision. (Since the collision is head-on, all motion is along a line.)

A crate of mass $M$ starts from rest at the top of a fric- tionless ramp inclined at an angle $\alpha$ above the horizontal. Find its speed at the bottom of the ramp, a distance $d$ from where it started. Do this in two ways: Take the level at which the potential energy is zero to be
(b) at the top of the ramp with $y$ positive upward.

The system shown in Figure P5.83 consists of a light, inextensible cord, light frictionless pulleys, and blocks of equal mass. Initially, the blocks are at rest the same height above the ground. The blocks are then released. Find the speed of block A at the moment when the vertical separation of the blocks is $h$.

A cylindrical piece of 8660 steel is to be austenitized and quenched in moderately agitated oil. If the hardness at the surface of the piece must be at least 58 HRC, what is the maximum allowable diameter? Justify your answer.

In an experiment, one of the forces exerted on a proton is $\vec{\boldsymbol{F}}=-\alpha x^{2} \hat{\imath}$ where $\alpha=12 \mathrm{N} / \mathrm{m}^{2}$. (a) How much work does $\vec{\boldsymbol{F}}$ do when the proton moves along the straight-line path from the point (0.10 m, 0) to the point (0.10 m, 0.40 m)? (b) Along the straight-line path from the point (0.10 m, 0) to the point (0.30 m, 0)? (c) Along the straight-line path from the point (0.30 m, 0) to the point (0.10 m, 0)? (d) Is the force $\vec{\boldsymbol{F}}$ conservative? Explain. If $\vec{\boldsymbol{F}}$ is conservative, what is the potential-energy function for it? Let U = 0 when x = 0.

Find the initial and final values of the time function f(t) if $\textbf{F}$(s) is given as: a) $\mathbf{F}(s)=\frac{10(s+2)}{(s+1)(s+4)}$ b) $\mathbf{F}(s)=\frac{s^{2}+2 s+2}{(s+6)\left(s^{3}+4 s^{2}+8 s+4\right)}$ c) $\frac{2 s}{s^{2}+2 s+3}$

If there is a net nonzero force on a moving object, can the total work done on the object be zero? Explain, using an example.

Hydroelectric energy comes from gravity pulling down water through dams in rivers. Explain how such energy is really just a form of solar energy by tracing how the sun's energy is able to get the water from the ocean to the reservoir behind the dam.