Question

(a) In Problem 3, what is the speed of the book when it reaches the 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 (a) increase, decrease, or remain the same?

Short answer

1. The speed of the book when it reachesthehand $v_f=12.9\text{m/s}$
2. If we substituted a second book with twice the mass, thenthespeed of the book_{$v_f=12.9\text{m/s}$}
3. If, instead, the book were thrown down, the answer to final velocity would increase.

Step-by-step solution

Step by Step Solution

i) Mass of book_{$m=2\text{\hspace{0.17em}kg}$}

ii) Height from the ground _{$\left(D\right)=10\text{m}$}

iii) Stretched hand at distance _{$\left(d\right)=1.50\text{m}$}

iv) Gravitational acceleration $\left(D\right)=9.80{\text{m/s}}^{\text{2}}$

To understand the concept

The mechanical energy _{$E_{mec}$} of a system is the sum of its kinetic energyand potential energy.

Formula:

i)

ii)

iii)

Calculate the height

$h=D-d$

$h=10-1.50$

$h=8.50\text{m}$

(a) Calculate the speed of the book when it reaches the hands

From problem 3

_{$U_i=mgD$}

_{$\Rightarrow U_i=2\times 9.80\times 10$}

_{$\Rightarrow U_i=196\text{J}$}

_{$\Rightarrow U_f=mgd$}

_{$\Rightarrow U_f=2\times 9.80\times 1.5$}

_{$\Rightarrow U_f=29\text{\hspace{0.17em}J}$}

By using law of conservation of mechanical energy,

_{$K_i+U_i=K_f+U_f$}

_{$\Rightarrow 0+196=K_f+29$}

_{$\begin{array}{c}{K}_f=196-29\\ =167\text{\hspace{0.17em}J}\end{array}$}

$\Rightarrow K_f=\frac{1}{2}m{v}_f^2$

_{$\Rightarrow v_f=\sqrt{\frac{2K_f}{m}}$}

_{$\Rightarrow v_f=\sqrt{\frac{2\times 167}{2}}$}

_{$\Rightarrow v_f=\sqrt{167}$}

_{$\Rightarrow v_f=12.9\text{m/s}$}

(b) Calculate the speed if we substituted a second book with twice the mass

If mass is doubled, then$K_f$is also doubled.

$K_f=2\times 167$

$K_f=m{v}_f^2$

$\Rightarrow v_f=\sqrt{\frac{2K_f}{2m}}$

$\Rightarrow v_f=\sqrt{\frac{2\times 2\times 167}{2\times 2}}$

$\Rightarrow v_f=\sqrt{167}$

$\Rightarrow v_f=12.9\text{m/s}$

(c) Figure out whether the answer to (a) would increase, decrease, or remain the same ifthe book were thrown down

If$K_i\ne 0,$we find$K_f=mgh+K_i$ . The value of$K_i$is always positive. This would result in a larger value for$K_f$than in the previous parts, and thus it leads to a larger value for_{$v_f$}.