To find the magnitude of the velocity, we can use the law of cosines. The law of cosines states that in a triangle, the square of one side (c) is equal to the sum of the squares of the other two sides (a and b), minus twice the product of the two sides (a and b) multiplied by the cosine of the included angle (θ).
In this case, we have a triangle with sides a, b, and c, and angles θ1 and θ2.
Given:
Sum of velocities (c) = 4.5 m/s
Angle θ1 = 32.4°
Angle θ2 = 22.4°
We can write the equation for the law of cosines as follows:
c² = a² + b² - 2ab * cos(θ)
We want to find the magnitude of the velocity, which is equivalent to the length of side c.
Plugging in the given values, we have:
(4.5 m/s)² = a² + b² - 2ab * cos(θ)
Simplifying and rearranging the equation, we have:
a² + b² - 2ab * cos(θ) = 20.25
Now, we need to find the values of a and b. We can use trigonometric relationships to express a and b in terms of the given angles and the magnitude of the velocity.
a = c * cos(θ1)
b = c * cos(θ2)
Plugging in the given values, we have:
a = 4.5 m/s * cos(32.4°)
b = 4.5 m/s * cos(22.4°)
Now, substitute the values of a and b back into the rearranged equation:
(4.5 m/s * cos(32.4°))² + (4.5 m/s * cos(22.4°))² - 2 * (4.5 m/s * cos(32.4°)) * (4.5 m/s * cos(22.4°)) * cos(θ) = 20.25
Simplify the equation and solve for cos(θ):
cos(θ) = (20.25 - (4.5 m/s * cos(32.4°))² - (4.5 m/s * cos(22.4°))²) / (2 * (4.5 m/s * cos(32.4°)) * (4.5 m/s * cos(22.4°)))
Calculate the value of cos(θ) using a calculator.
Finally, calculate the magnitude of the velocity by taking the square root of both sides of the equation:
velocity = √(a² + b² - 2ab * cos(θ))
Evaluate the expression to find the numeric value of the magnitude of velocity in meters per second.