In the figure shown below, a single frictionless roller-coaster car of mass $m=825 \mathrm{kg}$ tops the first hill with speed $v_0=20.0 \mathrm{m} {\mathrm{s}}^{-1}$ at height $h=50.0 \mathrm{m}.$

  

 

 

What is the speed of the car at (a) point $A$, (b) point $B$, and (c) point $C$? (d) How high will the car travel on the last hill, which is too high for it to cross? (e) If we substituted a second car with twice the mass. What then are the answers to parts (a) through (d)?

You drop a $2.00 \mathrm{kg}$ book to a friend who stands on the ground at distance $D=10.0 \mathrm{m}$ below. Your friend's outstretched hands are at a distance $d=1.50 \mathrm{m}$ above the ground (In the figure shown below).

(a) What is the speed of the book when it reaches your friend's hands?

(b) If we substituted a second book with twice the mass, what would its speed be?

(c) If, instead, the book were thrown down, would the answer to part (a) increase, decrease, or remain the same?

In the figure shown below, a $2.00 \mathrm{g}$ ice flake is released from the edge of a hemispherical bowl whose radius $r$ is $22.0 \mathrm{cm}.$ The flake-bowl contact is frictionless.

(a) What is the speed of the flake when it reaches the bottom of the bowl?

(b) If we substituted a second flake with twice the mass, what would its speed be?

(c) If, instead, we gave the flake an initial downward speed along the bowl, would the answer to part (a) increase, decrease, or remain the same?

A $1.50 \mathrm{kg}$ snowball is fired from a cliff $11.5 \mathrm{m}$ high. The snowball's initial velocity is $16.0 \mathrm{m} {\mathrm{s}}^{-1}$, directed $41.0°$ above the horizontal.

(a)Using energy techniques, find the speed of the snowball as it reaches the ground below the cliff. What is that speed?  

(b) if the launch angle is changed to $41.0°$ below the horizontal and (c) if the mass is changed to $3.00 \mathrm{kg}$?

The figure shown below shows a ball with mass $m=0.382 \mathrm{kg}$ attached to the end of a thin rod with length $L=0.498 \mathrm{m}$ and negligible mass. The other end of the rod is pivoted so that the ball can move in a vertical circle. The rod is held horizontally as shown and then given enough of a downward push to cause the ball to swing down and around and just reach the vertically up position, with zero speed there. How much work is done on the ball by the gravitational force from the initial point to,

(a) What initial speed must be given to the ball so that it reaches the vertically upward position with zero speed? What then is its speed at (b) the lowest point and (c) the point on the right at which the ball is level with the initial point? (d) If the ball's mass were doubled, would the answers to (a) through (c) increase, decrease, or remain the same?

In the figure shown below, a runaway truck with failed brakes is moving downgrade at $130 \mathrm{km} {\mathrm{h}}^{-1}$ just before the driver steers the truck up a frictionless emergency escape ramp with an inclination of $\mathrm{\theta }=15°.$ The truck's mass is $1.2\times {10}^4 \mathrm{kg}.$ (a) What minimum length $L$ must the ramp have, if the truck is to stop (momentarily) along with it? (Assume the truck is a particle and justify that assumption.) Does the minimum length $L$ increase, decrease, or remain the same if (b) the truck's mass is decreased and (c) its speed is decreased?

The figure shown below shows a thin rod of length $L=2.00 \mathrm{m}$ and negligible mass, that can pivot about one end to rotate in a vertical circle. A ball of mass $m=5.00 \mathrm{kg}$ is attached to the other end. The rod is pulled aside to angle ${\mathrm{\theta }}_0=30.0°$ and released with an initial velocity ${\overrightarrow{v}}_0=0.$ As the ball descends to its lowest point,

(a) What is the speed of the ball at the lowest point?

(b) Does the speed increase, decrease, or remain the same if the mass is increased?