91Ó°ÊÓ

A proton in an accelerator attains a speed of 0.88c. What

is the magnitude of the momentum of the proton?

An electron with a speed of 0.95c is emitted by a supernova, where c is the speed of light. What is the magnitude of the momentum of this electron?

A "cosmic-ray" proton hits the upper atmosphere with a speed 0.9999c, where c is the speed of light. What is the magnitude of the momentum of this proton? Note that $\left|\stackrel{→}{v}\right|\mathbf{/}\mathit{c}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9999}$; you don't actually need to calculate the speed$\left|v\right|$.

An object in the shape of a thin ring has radius a and mass M. A uniform sphere with mass m and radius R is placed with its center at a distance x to the right of the center of the ring, along a line through the center of the ring, and perpendicular to its plane (see Fig. E13.35). What is the gravitational force that the sphere exerts on the ring-shaped object? Show that your result reduces to the expected result when x is much larger than a.

If and , calculate the following quantities and make the requested comparisons: (a) ${\stackrel{→}{F}}_1+{\stackrel{→}{F}}_2$(b) $\left|{\stackrel{→}{F}}_1+{\stackrel{→}{F}}_2\right|$(c) $\left|{\stackrel{→}{F}}_1\right|+\left|{\stackrel{→}{F}}_2\right|$(d) Is $\left|{\stackrel{→}{F}}_1+{\stackrel{→}{F}}_2\right|=\left|{\stackrel{→}{F}}_1\right|+\left|{\stackrel{→}{F}}_2\right|$(e) ${\stackrel{→}{F}}_1-{\stackrel{→}{F}}_2$(f) $\left|{\stackrel{→}{F}}_1-{\stackrel{→}{F}}_2\right|$(g) $\left|{\stackrel{→}{F}}_1\right|-\left|{\stackrel{→}{F}}_2\right|$(h) Is$\left|{\stackrel{→}{F}}_1-{\stackrel{→}{F}}_2\right|=\left|{\stackrel{→}{F}}_1\right|-\left|{\stackrel{→}{F}}_2\right|$?

If $\stackrel{\mathbf{â‡\P Ä}}{{\mathbf{F}}_{\mathbf{1}}}\mathbf{=}<300,0,-200>$and , calculate the following

If and , calculate the following:(a) $$\begin{gathered} \stackrel{\mathbf{â‡\P Ä}}{{\mathbf{F}}_{\mathbf{1}}}\mathbf{+}\stackrel{\mathbf{â‡\P Ä}}{{\mathbf{F}}_{\mathbf{2}}}\mathbf{}\mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{}\left|\stackrel{â‡\P Ä}{F_1}+\stackrel{â‡\P Ä}{F_2}\right|\mathbf{}\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{}\left|\stackrel{â‡\P Ä}{F_1}+\stackrel{â‡\P Ä}{F_2}\right|\mathbf{}\mathbf{\left(}\mathit{d}\mathbf{\right)}\mathbf{}\left|\stackrel{â‡\P Ä}{F_1}+\stackrel{â‡\P Ä}{F_2}\right|\mathbf{=}\left|\stackrel{â‡\P Ä}{F_1}\right|\mathbf{+}\left|\stackrel{â‡\P Ä}{F_2}\right|\mathbf{?}\mathbf{}\mathbf{\left(}\mathit{e}\mathbf{\right)}\mathbf{}{\stackrel{\mathbf{â‡\P Ä}}{\mathbf{F}}}_{\mathbf{1}}\mathbf{-}\stackrel{\mathbf{â‡\P Ä}}{{\mathbf{F}}_{\mathbf{2}}}\mathbf{} \\ \mathbf{\left(}\mathit{f}\mathbf{\right)}\mathbf{}\left|\stackrel{â‡\P Ä}{F_1}-\stackrel{â‡\P Ä}{F_2}\right|\mathbf{}\mathbf{\left(}\mathit{g}\mathbf{\right)}\mathbf{}\mathbf{}\left|\stackrel{â‡\P Ä}{F_1}\right|\mathbf{-}\left|\stackrel{â‡\P Ä}{F_2}\right|\mathbf{}\mathbf{}\mathbf{\left(}\mathit{h}\mathbf{\right)}\mathbf{}\mathit{I}\mathit{s}\mathbf{}\left|\stackrel{â‡\P Ä}{F_1}-\stackrel{â‡\P Ä}{F_2}\right|\mathbf{=}\left|\stackrel{â‡\P Ä}{F_1}\right|\mathbf{-}\left|\stackrel{â‡\P Ä}{F_2}\right|\mathbf{?} \end{gathered}$$

Answer the following questions about the factor$\mathit{γ}$ (gamma) in the full relativistic equation for momentum:

(a) Is$\mathit{γ}$ a scalar or a vector quantity?

(b) What is the minimum possible value of $\mathit{γ}$?

(c) Does$\mathit{γ}$ reach its minimum value when an object’s speed is high or low?

(d) Is there a maximum possible value for $\mathit{γ}$?

(e) Does$\mathit{γ}$ become large when an object’s speed is high or low?

(f) Does the approximation$\mathit{γ}\mathbf{≈}\mathbf{1}$ apply when an object’s speed is low or when it is high?

Which of the following statements about the three vectors in Figure 1.24 are correct? $\mathbf{\left(}\mathit{a}\mathbf{\right)}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{s}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{t}}\mathbf{-}\stackrel{\mathbf{→}}{\mathbf{r}}\mathbf{}\mathbf{\left(}\mathit{b}\mathbf{\right)}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{r}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{t}}\mathbf{-}\stackrel{\mathbf{→}}{\mathbf{s}}\mathbf{}\mathbf{\left(}\mathit{c}\mathbf{\right)}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{r}}\mathbf{+}\stackrel{\mathbf{→}}{\mathbf{t}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{s}}\mathbf{}\mathbf{\left(}\mathit{d}\mathbf{\right)}\mathbf{}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{s}}\mathbf{+}\stackrel{\mathbf{→}}{\mathbf{t}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{r}}\mathbf{\left(}\mathit{e}\mathbf{\right)}\mathbf{}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{r}}\mathbf{+}\stackrel{\mathbf{→}}{\mathbf{s}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{t}}$

In which of these situations is it reasonable to use the approximate equation for the momentum of an object, instead of the full relativistically correct equation? (1) A car traveling on an interstate highway (2) A commercial jet airliner flying between New York and Seattle (3) A neutron traveling at 2700 meters per second (4) A proton in outer space traveling at $\mathbf{2}\mathbf{×}{\mathbf{10}}^{\mathbf{8}}$ m/s (5) An electron in a television tube traveling$\mathbf{3}\mathbf{×}{\mathbf{10}}^{\mathbf{6}}$ m/s.

At 10:00 AM you are at the locationm. By 10:02 AM you have walked to locationm. (a) What is $∆\stackrel{→}{r}$, the change in your position? (b) What is$∆t$ , the time interval during which your position changed?