A boat is traveling at [tex]$15.0 \, m/s$[/tex] in the [tex]y[/tex]-direction.

A current is flowing at [tex]$4.00 \, m/s$[/tex] in the [tex]x[/tex]-direction.

What is the magnitude of the velocity of the boat?

[tex]v = \, ? \, m/s[/tex]



Answer :

To determine the magnitude of the resultant velocity of the boat, given its components in the x and y directions, we can use the Pythagorean theorem. Here's a detailed, step-by-step solution:

1. Identify the velocities in the x and y directions:
- The boat's velocity in the y-direction ([tex]\(v_y\)[/tex]) is [tex]\(15.0 \ \text{m/s}\)[/tex].
- The current's velocity in the x-direction ([tex]\(v_x\)[/tex]) is [tex]\(4.00 \ \text{m/s}\)[/tex].

2. Understand that these velocities form a right triangle:
- One leg of the triangle is the velocity in the y-direction [tex]\(v_y = 15.0 \ \text{m/s}\)[/tex].
- The other leg of the triangle is the velocity in the x-direction [tex]\(v_x = 4.00 \ \text{m/s}\)[/tex].

3. Apply the Pythagorean theorem to find the resultant velocity ([tex]\(v\)[/tex]):
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the resultant velocity, [tex]\(v\)[/tex]) is equal to the sum of the squares of the lengths of the other two sides (the velocities [tex]\(v_x\)[/tex] and [tex]\(v_y\)[/tex]).

Therefore:
[tex]\[ v^2 = v_x^2 + v_y^2 \][/tex]

4. Substitute the given values:
[tex]\[ v^2 = (4.00 \ \text{m/s})^2 + (15.0 \ \text{m/s})^2 \][/tex]

5. Perform the calculations:
[tex]\[ v^2 = 4.00^2 + 15.0^2 \][/tex]
[tex]\[ v^2 = 16 + 225 \][/tex]
[tex]\[ v^2 = 241 \][/tex]

6. Take the square root of both sides to solve for [tex]\(v\)[/tex]:
[tex]\[ v = \sqrt{241} \][/tex]

7. Find the numerical value:
[tex]\[ v \approx 15.524 \ \text{m/s} \][/tex]

Thus, the magnitude of the boat's velocity is approximately [tex]\( 15.524 \ \text{m/s} \)[/tex].