Suppose a body that is acted on by exactly two forces is ac- celerated. Does it then follow that $(a)$ the body cannot move with constant speed; $(b)$ the velocity can never be zero; $(c)$ the sum of the two forces cannot be zero; $(d)$ the two forces must act in the same line?

Why do tires grip the road better on level ground than they do when going uphill or downhill?

In a Broadway performance, an 80.0-kg actor swings from a 3.75-m-long cable that is horizontal when he starts. At the bottom of his arc, he picks up his 55.0-kg costar in an inelastic collision. What maximum height do they reach after their upward swing?

(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.

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.

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.

From equations we saw earlier to obtain the dispersion coefficient for matter waves (in vacuum), then show that probability density we saw earlier follows from we saw earlier.