Question

A 216-m-wide river flows due east at a uniform speed of 2.2 m/s. A boat with speed of 7.9 m/s relative to the water leaves the south bank of the river pointed in a direction 34 degrees west of north. What is the magnitude of the boat's velocity relative to the ground (in m/s)

Answer 1

Answer:

The magnitude of the velocity of boat w.r.t ground is 6.2015 m/s

Solution:

As per the question;

Width of the river, w = 216 m

Uniform speed of boat due east, $u_{x} = 2.2\hat{i} m/s$

Angle in north west direction, $\theta = 34^{\circ}$

Velocity of the boat relative to water, $v_{bw} = 7.9m/s$

Now,

Velocity of the boat relative to the water is given by:

$v_{bw} = -7.9cos34^{\circ}\hat{i} + 7.9sin34^{\circ}\hat{j}$

$v_{bw} = - 6.55\hat{i} + 4.42\hat{j} m/s$

Also, velocity of water w.r.t ground is $v_{wg} = u_{x} = 2.2\hat{i} m/s$

Now,

Magnitude of the velocity of boat w.r.t ground is given by:

$v_{bg} = v_{bw} + v_{wg}$

$v_{bg} = - 6.55\hat{i} + 4.42\hat{j} + 2.2\hat{i}$

$v_{bg} = - 4.35\hat{i} + 4.42\hat{j} m/s$

$|v_{bg}| = \sqrt{(- 4.35)^{2} + (4.42)^{2}} = 6.2015\ m/s$