Question

Huck Finn walks at a speed of 0.70 m/s across his raft (that is, he walks perpendicular to the raft’s motion relative to the shore). The heavy raft is traveling down the Mississippi River at a speed of 1.50 m/s relative to the river bank (Fig. 3–42). What is Huck’s velocity (speed and direction) relative to the river bank?

FIGURE 3-42Problem 39

Short answer

Huck’s velocity relative to the river bank is $1.65{\mathrm{ms}}^{-1}$in the direction that makes an angle of $25°$with the river bank.

Step-by-step solution

Step 1. Relative velocity

You can find the relative velocity of an object with respect to a frame of reference if its velocity with respect to another frame of reference and the relative velocity of the two reference frames are known. Take, for example, the following.

If an object A moves with velocity${\overrightarrow{v}}_{\mathrm{AB}}$with respect to a frame of reference B and object B moves with velocity${\overrightarrow{v}}_{\mathrm{BC}}$with respect to another frame of reference C, then the relative velocity of A with respect to C is given by:

${\overrightarrow{v}}_{\mathrm{AC}}={\overrightarrow{v}}_{\mathrm{AB}}+{\overrightarrow{v}}_{\mathrm{BC}}$

Step 2. Given information

The magnitude of the velocity of Huck Finn with respect to his raft is $v_{\mathrm{HR}}=0.70{\mathrm{ms}}^{-1}$.

The magnitude of the velocity of the raft with respect to the bank of the river is$v_{\mathrm{RB}}=1.50{\mathrm{ms}}^{-1}$.

Let the velocity of Huck Finn relative to the river bank be ${\overrightarrow{v}}_{\mathrm{HB}}$.

Step 3. Vector representation of the motion of Huck Finn

The velocity of Huck Finn relative to the river bank is equal to the vector sum of the velocity of Huck Finn with respect to his raft and the velocity of the raft with respect to the bank of the river, i.e.,

${\overrightarrow{v}}_{\mathrm{HB}}={\overrightarrow{v}}_{\mathrm{HR}}+{\overrightarrow{v}}_{\mathrm{RB}}$

Since the motion of Huck Finn is perpendicular to the raft motion with respect to the shore, his motion can be shown as:

Here,$\theta$ is the angle which the velocity vector of Huck Finn makes with respect to the river bank.

Step 4. Determination of the magnitude of Huck’s velocity relative to the riverbank

From the figure, the magnitude of the velocity of Huck Finn relative to the riverbank can be found using the Pythagoras theorem as:

$\begin{array}{c}{v}_{\mathrm{HB}}=\sqrt{v_{\mathrm{HR}}^2+v_{\mathrm{RB}}^2}\\ =\sqrt{{\left(0.70{\mathrm{ms}}^{-1}\right)}^2+{\left(1.50{\mathrm{ms}}^{-1}\right)}^2}\\ =\sqrt{2.74{\mathrm{ms}}^{-1}}\\ =1.65{\mathrm{ms}}^{-1}\end{array}$

Step 5. Determination of the direction of motion of Huck relative to the river bank

From the figure, angle $\theta$can be found as follows:

$\begin{array}{c}\tan \theta =\frac{v_{\mathrm{HR}}}{v_{\mathrm{RB}}}\\ =\frac{\left(0.70{\mathrm{ms}}^{-1}\right)}{\left(1.50{\mathrm{ms}}^{-1}\right)}\\ =0.467\\ \theta ={\tan }^{-1}\left(0.467\right)\\ =25°\end{array}$

Thus, the velocity of Huck Finn relative to the river bank is $1.65{\mathrm{ms}}^{-1}$in the direction that makes an angle of $25°$with the river bank.