Chapter 9.2, Problem 25PP

To determine

The speed and the direction of motion of canoe and Judi if their combined mass is $115\text{ kg}$ .

Answer to Problem 25PP

  $v_{Jf}=2.78\text{ m/s}$ , in the opposite direction.

Explanation of Solution

Given:

Mass of the Carmen is $m_C=80.0\text{ kg}$

After leaving the canoe, Carmen moves with a speed of $v_{Cf}=4.0\text{ m/s}$ (final velocity of Carmen)

Judi is still in the canoe and it moves.

Combined mass of Judi and canoe is ${\text{m}}_J=115\text{ kg}$

Formula used:

Since momentum is conserved, total initial momentum of a given system is equal to the total final momentum of that system. In this problem system comprised of Carmen and Judi. Hence the conservation of momentum equation can be represented as,

  $p_{Ci}+p_{Ji}=p_{Cf}+p_{Jf}$$\left(1\right)$

Where,

  $p_{Ci}$ and $p_{Ji}$ are initial momentum of Carmen and Judi respectively,

  $p_{Cf}$ and $p_{Jf}$ are final momentum of Carmen and Judi respectively,

Since momentum is the product of mass and velocity of a moving object $\left(p=mv\right)$ , equation $\left(1\right)$ can be written as,

  $m_C{v}_{Ci}+m_J{v}_{Ji}=m_C{v}_{Cf}+m_J{v}_{Jf}$$\left(2\right)$

Where,

  $m_C$ and $m_J$ are the mass of Carmen and Judi respectively,

  $v_{Ci}$ and $v_{Ji}$ are the initial velocity of Carmen and Judi respectively,

  $v_{Cf}$ and $v_{Jf}$ are the final velocity of Carmen and Judi respectively,

Since initial velocity of Carmen and Judi is zero, equation $\left(2\right)$ becomes,

  $m_C\left(0\right)+m_J\left(0\right)=m_C{v}_{Cf}+m_J{v}_{Jf}$

  $\Rightarrow m_C{v}_{Cf}+m_J{v}_{Jf}=0$

  $\Rightarrow m_J{v}_{Jf}=-m_C{v}_{Cf}$

  $\Rightarrow v_{Jf}=\frac{-m_C{v}_{Cf}}{m_J}$$\left(3\right)$

Calculation:

The speed and the direction of motion of canoe and Judi can be obtained by substituting the numerical values in equation $\left(3\right)$ ,

  $v_{Jf}=\frac{\left(-80.0\text{ kg}\right)\left(4.0\text{ m/s}\right)}{\left(115\text{ kg}\right)}$

  $=-2.78\text{ m/s}$

Or

  $v_{Jf}=2.78\text{ m/s}$ , in the opposite direction.

Conclusion:

The speed of motion of Judi and canoe is $2.78\text{ m/s}$ and they move in opposite direction.