he x-component of a velocity vector that has an angle of 44∘ to the +x-axis has a magnitude of 4.3m/s . What is the magnetude velocity?
X = 4.3 m/s.
A = 44o
V = X/cosA = 4.3/cos44 = 5.98 m/s =
Total velocity.
To find the magnitude of the velocity vector, you can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the x-component of the velocity vector and the y-component of the velocity vector form a right triangle. The x-component corresponds to the side adjacent to the angle, and the y-component corresponds to the side opposite the angle.
Given that the x-component has a magnitude of 4.3 m/s, we can define it as follows:
x-component = 4.3 m/s
Since we want to find the magnitude of the velocity vector, we need to find the y-component using the given angle.
The trigonometric function that relates the sides of a right triangle to an angle is the sine function, which is defined as:
sin(theta) = opposite/hypotenuse
In this case, the opposite side is the y-component of the velocity vector, and the hypotenuse is the magnitude of the velocity vector.
We know that the angle is 44 degrees and the hypotenuse is the magnitude of the velocity vector.
So, we can write the equation as:
sin(44°) = y-component / magnitude of velocity
Now, rearranging the equation to solve for the y-component:
y-component = sin(44°) * magnitude of velocity
Substituting the given information, we have:
y-component = sin(44°) * magnitude of velocity
To find the magnitude of the velocity vector, we can use the Pythagorean theorem:
magnitude of velocity = sqrt[(x-component)^2 + (y-component)^2]
Substituting the given values:
magnitude of velocity = sqrt[(4.3 m/s)^2 + (sin(44°) * magnitude of velocity)^2]
Now, we need to solve this equation. Let's calculate the value step by step.
Start by squaring both the x-component and y-component:
magnitude of velocity = sqrt[ (4.3 m/s)^2 + (sin(44°) * magnitude of velocity)^2 ]
Simplifying further:
magnitude of velocity = sqrt[ 18.49 m^2/s^2 + (sin(44°))^2 * magnitude of velocity^2 ]
Simplifying the equation:
magnitude of velocity = sqrt[ 18.49 m^2/s^2 + (0.6947)^2 * magnitude of velocity^2 ]
Now, square both sides to eliminate the square root:
magnitude of velocity^2 = 18.49 m^2/s^2 + (0.6947)^2 * magnitude of velocity^2
Combine like terms:
magnitude of velocity^2 - (0.6947)^2 * magnitude of velocity^2 = 18.49 m^2/s^2
Now, factor out the magnitude of velocity^2:
(1 - (0.6947)^2) * magnitude of velocity^2 = 18.49 m^2/s^2
Calculating the value inside the parentheses:
(1 - 0.4822) * magnitude of velocity^2 = 18.49 m^2/s^2
(0.5178) * magnitude of velocity^2 = 18.49 m^2/s^2
Now, divide both sides by 0.5178 to solve for the magnitude of velocity:
magnitude of velocity^2 = 18.49 m^2/s^2 / 0.5178
magnitude of velocity^2 ≈ 35.7 m^2/s^2
Finally, take the square root of both sides to find the magnitude of velocity:
magnitude of velocity ≈ sqrt(35.7 m^2/s^2)
magnitude of velocity ≈ 5.98 m/s
Therefore, the magnitude of the velocity vector is approximately 5.98 m/s.