Question

A boat is traveling upstream in the positive direction of an x axis at $\mathbf{14}\mathbf{}\mathit{k}\mathit{m}\mathbf{/}\mathit{h}$with respect to the water of a river. The water is flowing at $\mathbf{9}\mathbf{.}\mathbf{0}\mathbf{}\mathit{k}\mathit{m}\mathbf{/}\mathit{h}$with respect to the ground. What are the (a) magnitude and (b) direction of the boat’s velocity with respect to the ground? A child on the boat walks from front to rear at 6.0 km/h with respect to the boat. What are the (a) magnitude and (b) direction of the child’s velocity with respect to the ground?

Short answer

Answer

1. The magnitude of the boat’s velocity with respect to the ground$5km/h$

2. The direction of the boat’s velocity with respect to the ground is positive

3. The magnitude of the child’s velocity with respect to the ground$1km/h$

4. The direction of the child’s velocity with respect to the ground is negative

Step-by-step solution

Given data

1. The velocity of the boat relative to water is$v_{bw}=14km/h$

2. The velocity of water relative to the ground is$v_{wg}=-9.0km/h$

The velocity of the child relative to the boat is$v_{cb}=-6.0km/h$

Understanding the concept of relative motion

The term "relative motion" refers to the comparison of the relative accelerations and velocities of two rigid bodies. The boat is traveling along positive x-direction relative to water, and the water is flowing relative to the ground. Hence, according to the relative motion concept, we can find the boat’s velocity relative to the ground.

Formula:

$$\begin{gathered} v_{bg}=v_{bw}+v_{wg} \\ {v}_{cb}=v_{cb}+c_{bg} \end{gathered}$$

(a) Calculate the magnitude of the boat’s velocity with respect to ground

According to the equation of relative motion,

$$\begin{gathered} v_{bg}=v_{bw}+v_{wg} \\ =\left(14km/h\right)+\left(-9.0km/h\right) \\ =5.00km/h \end{gathered}$$

The magnitude of the boat’s velocity with respect to ground is $5.00km/h$.

(b) Calculate the direction of the boat’s velocity with respect to the ground

The direction of the boat’s velocity with respect to the ground is positive that is upstream.

(c) Calculate the magnitude of the child’s velocity with respect to the ground

According to the relative motion,

$$\begin{gathered} v_{cg}=v_{cb}+v_{bg} \\ =\left(-6.0km/h\right)+\left(5.0km/h\right) \\ =-1.0km/h \end{gathered}$$

So, the magnitude of the child’s velocity with respect to the ground is $1.0km/h$

(d) Calculate the direction of the child’s velocity with respect to the ground

The direction of the child’s velocity with respect to the ground$v_{cg}$ is negative that is downstream.