A piano is rolling down a frictionless slope at an ever increas- ing speed. The piano tuner sees it, runs up to it and pushes on it, slowing it down to a constant speed. The magnitude of the force on the man by the piano is $F_{MP}$ ; the magnitude of the force on the piano by the man is $F_{PM}$. If we compare these forces, we find $(A)$ $F_{PM}>F_{MP}$ always. $(B)$ $F_{PM}>F_{MP}$ while the piano slows down but $F_{PM}=$ $F_{MP}$ when the piano is moving at constant speed. $(C)$ $F_{PM}=F_{MP}$ always. $(D)$ $F_{PM}=F_{MP}$ while the piano slows down but $F_{PM}<$ $F_{MP}$ when the piano is moving at constant speed.

(a) Neglecting gravitational forces, what force would be re- quired to accelerate a $1200$-metric-ton spaceship from rest to one-tenth the speed of light in $3$ days? In $2$ months? (One metric ton $=1000$ kg.) (b) Assuming that the engines are shut down when this speed is reached, what would be the time required to complete a $5$-light-month journey for each of these two cases? (Use $1$ month $=30$ days.)

Two blocks are in contact on a frictionless table. A horizontal force is applied to one block, as shown in earlier Fig. $(a)$ If $m_1=2.3$ kg, $m_2=1.2$ kg, and $F=3.2$ N, find the force of contact between the two blocks. $(b)$ Show that if the same force $F$ is applied to $m_2$ rather than to $m_1$ , the force of contact between the blocks is $2.1$ N, which is not the same value de- rived in $(a)$. Explain.

What is the relation$—$if any$—$between the force acting on an object and the direction in which the object is moving?

A rock is on an inclined surface. The rock is originally at rest, but starts to slide down the incline. The magnitude of the force on the surface due to the rock is $F_{SR}$ , and the magnitude of the force on the rock due to the surface is $F_{RS}$. If we com- pare these forces, we find
$(A)$ $F_{SR}<F_{RS}$. always. $(B)$ $F_{SR}=F_{RS}$ when the rock is at rest but then $F_{SR}>F_{RS}$. $(C)$ $F_{SR}=F_{RS}$ always. $(D)$ $F_{SR}>F_{RS}$ always.

Of the objects listed in earlier Table, which might be considered as a particle for the motion described? For those that behave like particles, can you describe a type of motion in which they could $\it{not}$ be considered as particles?

A $10.0$ kg object is launched vertically into the air with an initial velocity of $50.0$ m/s. In addition to the force of gravity there is a frictional force which is proportional to velocity ac- cording to $f_y=-bv_y$; note that this frictional force is nega- tive (down) when the object is moving up, but positive (up) when the object is moving down. (a) Numerically generate distance-time graphs for the object, using $b=0$, but use several different step sizes for $\Delta$ $t$, such as $1.0$ s, $0.1$ s, $0.01$ s, and $0.001$ s. Show the results on a sin- gle graph. How does the highest point vary with the step size? (b) Numerically generate distance-time graphs for the object, using a step size of $\Delta$ $t=0.01$ s. Now, however, try non-zero values for $b$, such as $0.1~\mathrm{N}~\cdot$ s/m, $0.5~\mathrm{N}~\cdot$ s/m, $1.0~\mathrm{N}~\cdot$ s/m, $5.0~\mathrm{N}~\cdot$ s/m and $10.0~\mathrm{N}~\cdot$ s/m. How does the highest point vary with $b?$ What do you notice about the shape of the graphs as $b$ increases?

Suppose that the Sun’s gravitational force was suddenly turned off, so that Earth became a free object rather than be- ing confined to orbit the Sun. How long would it take for Earth to reach a distance from the Sun equal to Pluto’s pres- ent orbital radius? (Hint: You will find some of the data you need in earlier Appendix.)

An interstellar spacecraft, far from the influence of any stars or planets, is moving at high speed under the influence of fu- sion rockets when the engines malfunction and stop. The spacecraft will
$(A)$ immediately stop, throwing all of the occupants to the front of the craft. $(B)$ begin slowing down, eventually coming to a rest in the cold emptiness of space. $(C)$ keep moving at constant speed for a while, but then be- gin to slow down. $(D)$ keep moving forever at the same speed.

A ball rolls in a straight line along the horizontal direction. Using motion diagrams (or multiflash photographs), describe the velocity and acceleration of the ball for each of the following situations: (a) The ball moves to the right at a constant speed. (b) The ball moves from right to left and continually slows down. (c) The ball moves from right to left and continually speeds up. (d) The ball moves to the right, first speeding up at a constant rate and then slowing down at a constant rate.

A 71 kg baseball player jumps straight up to catch a hard-hit ball. If the 140 g ball is moving horizontally at 28 m/s, and the catch is made when the ballplayer is at the highest point of his leap, what is his speed immediately after stopping the ball?

An athlete throws a basketball upward from the ground, giving it speed $10.6 \mathrm{~m} / \mathrm{s}$ at an angle of $55.0^{\circ}$ above the horizontal. (a) What is the acceleration of the basketball at the highest point in its trajectory? (b) On its way down, the basketball hits the rim of the basket, $3.05 \mathrm{~m}$ above the floor. It bounces straight up with one-half the speed with which it hit the rim. What height above the floor does the basketball reach on this bounce?

A roller-coaster car at an amusement park has a mass of 1200 kg when fully loaded with passengers. As the car passes over the top of a circular hill of radius 18 m, assume that its speed is not changing. At the top of the hill, what are the (a) magnitude

$$F_N$$

and (b) direction (up or down) of the normal force on the car from the track if the car’s speed is ν = 11 m/s? What are (c)

$$F_N$$

and (d) the direction if ν = 14 m/s?

Trained dolphins are capable of a vertical leap of $7.0 \mathrm{~m}$ Blo straight up from the surface of the water-an impressive feat. NT Suppose you could train a dolphin to launch itself out of the water at this same speed but at an angle. What maximum horizontal range could the dolphin achieve?