Question

Shipis located 4.0 kmnorth and 2.5 kmeast of ship B. Ship Ahas a velocity of 22 km/htoward the south, and ship Bhas a velocity of 40 km/hin a direction$\mathbf{37}\mathbf{°}$ north of east. (a)What is the velocity of Arelative to Bin unit-vector notation with toward the east? (b)Write an expression (in terms of $\hat{\mathbf{i}}$and$\hat{\mathbf{j}}$) for the position ofArelative to Bas a function of t, wheret=0when the ships are in the positions described above. (c)At what time is the separation between the ships least? (d)What is that least separation?

Short answer

a) The magnitude of the velocity of ship A relative to ship B 56 km/hr

b) The expression for the position of A relative to B as a function of time is $\left(2.5-32\mathrm{t}\right)\hat{\mathrm{I}}+\left(4-46\mathrm{t}\right)\hat{\mathrm{j}}$.

c) The time at which the separation between two ships will be least is 0.084 hrs.

d) The least separation between the two ships is 0.2km.

Step-by-step solution

 Step 1: The given data

1. The ship A is located 4.0km north and 2.5 km east of ship B.
2. The velocity of A,$V_A=22\mathrm{km}/\mathrm{h}$ in south direction.
3. The velocity of B,${\mathrm{V}}_{\mathrm{B}}=40\mathrm{km}/\mathrm{h}$ in a direction

Understanding the concept of the relative motion

When two frames of reference Aand Bare moving relative to each other at a constant velocity, the velocity of a particle Pas measured by an observer in frame Ais different than the velocity measured in frame B. The velocities measured in two reference frames are related by,

${\overrightarrow{\mathbf{v}}}_{\mathbf{P}\mathbf{A}}\mathbf{=}{\overrightarrow{\mathbf{v}}}_{\mathbf{P}\mathbf{B}}\mathbf{+}{\overrightarrow{\mathbf{v}}}_{\mathbf{B}\mathbf{A}}$

Here${\overrightarrow{\mathbf{v}}}_{\mathbf{BA}}$ is the velocity of B with respect to A.

Using the relative motion concept, we can find the magnitude and the direction of the resultant vector. Using these magnitudes of the velocity vectors, we can find the required time and least separation between the ships.

Formulae:

The magnitude of the vector $\stackrel{\rightharpoonup }{V}$is given by, $V=\sqrt{V_x^2+V_y^2}$ (i)

The velocity vector in terms of unit-vector notation, $\stackrel{\rightharpoonup }{v}=v\cos \theta \hat{i}+v\sin \theta \hat{j}$ (ii)

(a) Calculation of the magnitude of the velocity of ship A relative to ship B

Vector diagram:

The velocity of A can be written in the vector notation as:

$\overrightarrow{V_A}=\left(-22\right)\hat{j}$

The velocity of B can be written in the vector notation using equation (ii) as:

$$\begin{gathered} \overrightarrow{V_B}=40\cos 37°\hat{i}+40\sin 37°\hat{j} \\ =31.95\hat{i}+24.07\hat{j} \end{gathered}$$

The velocity of A with respect to B is given as:

$$\begin{gathered} \overrightarrow{V_{AB}}=\overrightarrow{V_A}-\overrightarrow{V_B} \\ =\left(-22\right)\hat{j}-\left(31.95\hat{i}+24.07\hat{j}\right) \\ =-32\hat{i}-46\hat{j} \end{gathered}$$

Now, the magnitude of the velocity vector can be given using equation (i) as follows:

$$\begin{gathered} V_{AB}=\sqrt{{\left(-32\right)}^2+{\left(-46\right)}^2} \\ =56.04\mathrm{km}/\mathrm{hr} \\ \approx 56\mathrm{km}/\mathrm{hr} \end{gathered}$$

Hence, the value of the velocity is 56 km/hr.

(b) Calculation for the expression of position of A relative to B

The position of A with respect to B is given as follows:

$$\begin{gathered} {\overrightarrow{r}}_{AB}-{\overrightarrow{r}}_{0AB}=\int \overrightarrow{V_{AB}}dt \\ =\int \left(-32\hat{i}-46\hat{j}\right)dt \\ =\int \left(-32t\hat{i}-46t\hat{j}\right) \\ {\overrightarrow{r}}_{AB}=\left(2.5-32t\right)\hat{i}+\left(4-46t\right)\hat{j} \end{gathered}$$

Hence, the expression for the position is$\left(2.5-32t\right)\hat{i}+\left(4-46t\right)\hat{j}$ .

(c) Calculation of the time for the least amount of time

The separation between two boats will be minimum if$\frac{d{r}_{AB}}{dt}=0$

Thus, solving the expression for the position, we can get the time value as follows:

$$\begin{gathered} \frac{d}{dt}\left[\sqrt{{\left(2.5-32t\right)}^2{\left(4-46t\right)}^2}\right]=0 \\ \frac{6286t-528}{\sqrt[2]{{\left(2.5-32t\right)}^2{\left(4-46t\right)}^2}}=0 \\ 6286t=528 \\ t=0.08399 \\ \approx 0.084\mathrm{h} \end{gathered}$$

Hence, the value of the time is $0.084\mathrm{h}$.

d) Calculation of the minimum separation between the ships

The minimum separation between two boats at t=0.084 h is given as:

$$\begin{gathered} r_{AB}=\sqrt{{\left(2.5-32\left(0.084\right)\right)}^2+{\left(4-46\left(0.084\right)\right)}^2} \\ =0.225 \\ ~0.2\mathrm{km} \end{gathered}$$

Hence, the value of the separation is 0.2 km.