To determine the magnitude of the resultant velocity given [tex]\( v_x = 3 \)[/tex] meters/second and [tex]\( v_y = 4 \)[/tex] meters/second, we use the Pythagorean theorem. Here's the step-by-step solution:
1. Understand that [tex]\( v_x \)[/tex] and [tex]\( v_y \)[/tex] represent the components of the velocity in the x and y directions respectively.
2. To find the magnitude of the resultant velocity [tex]\( v \)[/tex], we treat the situation as a right triangle where [tex]\( v_x \)[/tex] and [tex]\( v_y \)[/tex] are the legs of the triangle. The magnitude of the resultant velocity [tex]\( v \)[/tex] is the hypotenuse of this right triangle.
3. Apply the Pythagorean theorem:
[tex]\[
v = \sqrt{v_x^2 + v_y^2}
\][/tex]
4. Substitute the given values [tex]\( v_x = 3 \)[/tex] and [tex]\( v_y = 4 \)[/tex]:
[tex]\[
v = \sqrt{3^2 + 4^2}
\][/tex]
5. Calculate the squares of the components:
[tex]\[
v = \sqrt{9 + 16}
\][/tex]
6. Add the squares:
[tex]\[
v = \sqrt{25}
\][/tex]
7. Take the square root of 25:
[tex]\[
v = 5
\][/tex]
Thus, the magnitude of the resultant velocity is 5 meters/second.
Now, match this result with the given options:
A. 7 meters/second
B. 5 meters/second
C. 1 meters/second
D. 6 meters/second
The correct answer is:
B. 5 meters/second
