answer:Given: Table height, {eq}h = 1.225 m{/eq} Distance from base, {eq}x = 0.400 m{/eq} Gravity, {eq}g = 9.81 m/s^2{/eq} a) To find the time of fall, we can use the kinematic equation for vertical motion under gravity: {eq}h = frac{1}{2} g t^2{/eq} Solving for time {eq}t{/eq}: {eq}t = sqrt{frac{2h}{g}}{/eq} {eq}t approx sqrt{frac{2 times 1.225}{9.81}}{/eq} {eq}t approx 0.5 s{/eq} The car takes about 0.5 seconds to fall to the ground. b) Since the car moves horizontally without any acceleration (assuming no air resistance), its initial horizontal velocity {eq}u_x{/eq} can be calculated using the distance traveled and time: {eq}x = u_x t{/eq} Solving for {eq}u_x{/eq}: {eq}u_x = frac{x}{t}{/eq} {eq}u_x = frac{0.400}{0.5}{/eq} {eq}u_x = 0.8 m/s{/eq} Therefore, the speed of the car as it rolled off the table was 0.8 meters per second.

answer:The required coefficient of friction is given by: {eq}begin{align*} mu &= frac{r gtan (theta) - v^2 }{tan (theta) v^2 + r g }[0.2cm] &= rm frac{(50.0 m)(9.80 m/s^2)tan (20.0^o) - (8.33 m/s)^2 }{tan (20.0^o)(8.33 m/s)^2 + (50.0 m)(9.80 m/s^2) }[0.2cm] &= boxed{rm 0.176} end{align*} {/eq}

answer:We integrate the derivative vector function component-wise: {eq}begin{align*} mathbf r(t) &= int mathbf r'(t) , dt &= int (t^2 mathbf i + e^{-t} mathbf j + 2te^{2t} mathbf k) , dt &= left( int t^2 , dt right) mathbf i + left( int e^{-t} , dt right) mathbf j + left( int 2te^{2t} , dt right) mathbf k &= left( frac{t^3}{3} right) mathbf i + left( -e^{-t} right) mathbf j + left( te^{2t} - frac{e^{2t}}{2} right) mathbf k + C_1 mathbf i + C_2 mathbf j + C_3 mathbf k &= left( frac{t^3}{3} + C_1 right) mathbf i + left( -e^{-t} + C_2 right) mathbf j + left( te^{2t} - frac{e^{2t}}{2} + C_3 right) mathbf k end{align*} {/eq} Using the initial condition {eq}mathbf r(0) = mathbf i + mathbf j + mathbf k {/eq}, we find the constants of integration: {eq}begin{align*} mathbf r(0) &= left( frac{0^3}{3} + C_1 right) mathbf i + left( -e^{-0} + C_2 right) mathbf j + left( 0e^{2 cdot 0} - frac{e^{2 cdot 0}}{2} + C_3 right) mathbf k Rightarrow C_1 &= 1, C_2 = 2, C_3 = frac{3}{2} end{align*} {/eq} Thus, the vector function is: {eq}mathbf r(t) = left( frac{t^3}{3} + 1 right) mathbf i + left( -e^{-t} + 2 right) mathbf j + left( te^{2t} - frac{e^{2t}}{2} + frac{3}{2} right) mathbf k {/eq}

answer:Quindlen's metaphor of the "quilt" effectively conveys the central idea that the United States is a nation composed of diverse cultures and experiences. Like a quilt, which is made up of individual patches sewn together, the country is a tapestry of different perspectives and backgrounds. The metaphor highlights both the uniqueness of each "patch" and the interconnectedness that binds them together, emphasizing the idea that despite cultural differences, Americans can unite in times of crisis or shared experiences.