Chapter 1, Problem 27Q

To determine

The rank of arrows according to the maximum height that each achieves from greatest to smallest.

Answer to Problem 27Q

The rank of all arrows,

$\begin{array}{ccc}\text{Arrow}\text{A:}& M=0.075\text{kg}& V=16\text{m/s}\\ \text{Arrow}\text{B:}& M=0.180\text{kg}& V=12\text{m/s}\\ \text{Arrow}\text{C:}& M=0.100\text{kg}& V=18\text{m/s}\\ \text{Arrow}\text{D:}& M=0.075\text{kg}& V=12\text{m/s}\\ \text{Arrow}\text{E:}& M=0.120\text{kg}& V=10\text{m/s}\\ \text{Arrow}\text{F:}& M=0.090\text{kg}& V=16\text{m/s}\\ \text{Arrow}\text{G:}& M=0.180\text{kg}& V=10\text{m/s}\\ \text{Arrow}\text{H:}& M=0.050\text{kg}& V=8\text{m/s}\end{array}\begin{array}{c}\Rightarrow \text{Rank}\text{2}\\ \Rightarrow \text{Rank}\text{3}\\ \Rightarrow \text{Rank}\text{1}\\ \Rightarrow \text{Rank}\text{3}\\ \Rightarrow \text{Rank}4\\ \Rightarrow \text{Rank}\text{2}\\ \Rightarrow \text{Rank}\text{4}\\ \Rightarrow \text{Rank}\text{5}\end{array}$.

Explanation of Solution

Given information:

Eight arrows are given,

$\begin{array}{ccc}\text{Arrow}\text{A:}& M=0.075\text{kg}& V=16\text{m/s}\\ \text{Arrow}\text{B:}& M=0.180\text{kg}& V=12\text{m/s}\\ \text{Arrow}\text{C:}& M=0.100\text{kg}& V=18\text{m/s}\\ \text{Arrow}\text{D:}& M=0.075\text{kg}& V=12\text{m/s}\\ \text{Arrow}\text{E:}& M=0.120\text{kg}& V=10\text{m/s}\\ \text{Arrow}\text{F:}& M=0.090\text{kg}& V=16\text{m/s}\\ \text{Arrow}\text{G:}& M=0.180\text{kg}& V=10\text{m/s}\\ \text{Arrow}\text{H:}& M=0.050\text{kg}& V=8\text{m/s}\end{array}$

Concept Used:

Potential energy,

$E_{\text{p}}=mgh$.

Kinetic energy,

$E_{\text{k}}=\frac{1}{2}m{v}^2$.

Calculation:

At ground, potential energy is zero, since the height is zero at ground.

At the highest point, the velocity become zero due to the deceleration affected by the gravity. So, the kinetic energy will be zero at highest point.

According to the law of conservation of energy, the kinetic energy of arrow is completely converted into the potential energy at the highest point. So,

$\begin{array}{l}mgh=\frac{1}{2}m{v}^2\\ \Rightarrow h\propto v^2\end{array}$

The height is directly proportional to the square of initial velocity.

So, the order arrows based on maximum height is same as the order based on initial velocity.

Thus, the rank for each arrow is,

$\begin{array}{ccc}\text{Arrow}\text{A:}& M=0.075\text{kg}& V=16\text{m/s}\\ \text{Arrow}\text{B:}& M=0.180\text{kg}& V=12\text{m/s}\\ \text{Arrow}\text{C:}& M=0.100\text{kg}& V=18\text{m/s}\\ \text{Arrow}\text{D:}& M=0.075\text{kg}& V=12\text{m/s}\\ \text{Arrow}\text{E:}& M=0.120\text{kg}& V=10\text{m/s}\\ \text{Arrow}\text{F:}& M=0.090\text{kg}& V=16\text{m/s}\\ \text{Arrow}\text{G:}& M=0.180\text{kg}& V=10\text{m/s}\\ \text{Arrow}\text{H:}& M=0.050\text{kg}& V=8\text{m/s}\end{array}\begin{array}{c}\Rightarrow \text{Rank}\text{2}\\ \Rightarrow \text{Rank}\text{3}\\ \Rightarrow \text{Rank}\text{1}\\ \Rightarrow \text{Rank}\text{3}\\ \Rightarrow \text{Rank}4\\ \Rightarrow \text{Rank}\text{2}\\ \Rightarrow \text{Rank}\text{4}\\ \Rightarrow \text{Rank}\text{5}\end{array}$.

Conclusion:

The rank of all arrows,

$\begin{array}{ccc}\text{Arrow}\text{A:}& M=0.075\text{kg}& V=16\text{m/s}\\ \text{Arrow}\text{B:}& M=0.180\text{kg}& V=12\text{m/s}\\ \text{Arrow}\text{C:}& M=0.100\text{kg}& V=18\text{m/s}\\ \text{Arrow}\text{D:}& M=0.075\text{kg}& V=12\text{m/s}\\ \text{Arrow}\text{E:}& M=0.120\text{kg}& V=10\text{m/s}\\ \text{Arrow}\text{F:}& M=0.090\text{kg}& V=16\text{m/s}\\ \text{Arrow}\text{G:}& M=0.180\text{kg}& V=10\text{m/s}\\ \text{Arrow}\text{H:}& M=0.050\text{kg}& V=8\text{m/s}\end{array}\begin{array}{c}\Rightarrow \text{Rank}\text{2}\\ \Rightarrow \text{Rank}\text{3}\\ \Rightarrow \text{Rank}\text{1}\\ \Rightarrow \text{Rank}\text{3}\\ \Rightarrow \text{Rank}4\\ \Rightarrow \text{Rank}\text{2}\\ \Rightarrow \text{Rank}\text{4}\\ \Rightarrow \text{Rank}\text{5}\end{array}$.