Relative Velocity worksheet - Free Printable

Problem Analysis:

The problem involves the concept of relative velocity in a scenario where a boat is moving across a river. The river has a current with a known velocity, and the boat has its own velocity relative to the water. We need to determine the resultant velocity of the boat (relative velocity) and its direction.

#### Given:
- Velocity of the river (\( v_{\text{river}} \)): \( 1.2 \, \text{m/s} \) (in the positive \( x \)-direction).
- Velocity of the boat relative to the water (\( v_{\text{boat}} \)): \( 0.75 \, \text{m/s} \) (perpendicular to the river flow, i.e., in the positive \( y \)-direction).

#### Objective:
1. Calculate the magnitude of the resultant velocity (\( v_{\text{resultant}} \)).
2. Determine the direction of the resultant velocity relative to the positive \( x \)-axis.

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Step-by-Step Solution:

#### 1. Represent Velocities as Vectors:
- The velocity of the river (\( v_{\text{river}} \)) can be represented as:
\[
\vec{v}_{\text{river}} = 1.2 \, \hat{i} \, \text{m/s}
\]
(where \( \hat{i} \) is the unit vector in the \( x \)-direction).

- The velocity of the boat relative to the water (\( v_{\text{boat}} \)) can be represented as:
\[
\vec{v}_{\text{boat}} = 0.75 \, \hat{j} \, \text{m/s}
\]
(where \( \hat{j} \) is the unit vector in the \( y \)-direction).

#### 2. Calculate the Resultant Velocity:
The resultant velocity (\( \vec{v}_{\text{resultant}} \)) is the vector sum of \( \vec{v}_{\text{river}} \) and \( \vec{v}_{\text{boat}} \):
\[
\vec{v}_{\text{resultant}} = \vec{v}_{\text{river}} + \vec{v}_{\text{boat}}
\]
Substituting the given values:
\[
\vec{v}_{\text{resultant}} = 1.2 \, \hat{i} + 0.75 \, \hat{j}
\]

#### 3. Magnitude of the Resultant Velocity:
The magnitude of the resultant velocity is given by:
\[
v_{\text{resultant}} = \sqrt{(v_x)^2 + (v_y)^2}
\]
where \( v_x = 1.2 \, \text{m/s} \) and \( v_y = 0.75 \, \text{m/s} \). Substituting these values:
\[
v_{\text{resultant}} = \sqrt{(1.2)^2 + (0.75)^2} = \sqrt{1.44 + 0.5625} = \sqrt{2.0025} \approx 1.415 \, \text{m/s}
\]

#### 4. Direction of the Resultant Velocity:
The direction of the resultant velocity is given by the angle \( \theta \) it makes with the positive \( x \)-axis. This angle can be calculated using the tangent function:
\[
\tan \theta = \frac{v_y}{v_x}
\]
Substituting the values of \( v_y \) and \( v_x \):
\[
\tan \theta = \frac{0.75}{1.2} = 0.625
\]
Taking the inverse tangent:
\[
\theta = \tan^{-1}(0.625) \approx 32.01^\circ
\]
Thus, the direction is approximately \( 32^\circ \) above the positive \( x \)-axis.

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Final Answers:

(A) The relative velocity of the boat is:
\[
\boxed{1.42 \, \text{m/s}}
\]

(B) The direction is:
\[
\boxed{32^\circ}
\]