Chapter 8, Problem 1OQ

To determine

The ratio of the mass of the bean to the mass of the pebble.

Answer to Problem 1OQ

Option (a) , the ratio of the mass of the bean to mass of pebble is $\frac{1}{9}$.

Explanation of Solution

The kinetic energy  provided by slingshot is  same for the pebble as well as to the bean.

Write the expression for kinetic energy of pebble.

  $K_p=\frac{1}{2}{m}_p{v}_p^2$                                                                                                   (I)

Here, $K_p$ is Kinetic energy for pebble , $m_p$ is mass of pebble and $v_p$ is velocity of pebble.

Write the expression for kinetic energy of bean as.

  $K_b=\frac{1}{2}{m}_b{v}_b^2$                                                                                                   (II)

Here, $K_b$ is Kinetic energy for bean , $m_b$ is mass of bean and $v_b$ is velocity of bean.

Write the expression for the  conservation of kinetic energy as.

  $K_p=K_b$                                                                                                      (III)

Conclusion:

Substitute $\frac{1}{2}{m}_p{v}_p^2$ for $K_p$  and  $\frac{1}{2}{m}_b{v}_b^2$ for $K_b$ in equation (III).

    $\frac{1}{2}{m}_p{v}_p^2=\frac{1}{2}{m}_b{v}_b^2$

Rearrange the above equation for $\left(\frac{m_b}{m_p}\right)$ as.

  $\frac{m_b}{m_p}={\left(\frac{v_p}{v_b}\right)}^2$

Substitute $200\text{cm}/\text{s}$ for $v_p$ and $600\text{cm}/\text{s}$ for $v_b$ in the above equation.

  $\begin{array}{c}\frac{m_b}{m_p}={\left(\frac{200}{600}\right)}^2\\ =\frac{1}{9}\end{array}$

Thus, the ratio of the mass of the bean to mass of pebble is $\frac{1}{9}$, option (a) is correct.

Theratio of the mass of the bean to mass of pebble is equal to the square of the ratio of velocity of pebble to velocity of bean. Thus, option (a) is correct.

The ratio of the mass of the bean to the mass of pebble is not equal to the ratio of velocity of pebble to velocity of bean. Thus, option (b) is incorrect.

The value of velocity of pebble is not equal to  velocity of bean. Thus, option (c) is incorrect.

Theratio of the mass of the bean to mass of pebble is not equal to ratio of velocity of bean to velocity of  pebble . Thus, option (d) is incorrect.

Theratio of the mass of the bean to mass of pebble is not equal to the square of the  ratio of velocity of bean to velocity of  pebble. Thus, option (e) is incorrect.