Velocity a 30m/s at an angle of 45 degree to the horizontal and vector b 40 m/s at angle of 80 degree use the parallelogram rule to determine the resultant velocity

To find the resultant velocity using the parallelogram rule, we first need to break down each vector into its horizontal and vertical components. This will allow us to visualize and calculate the resultant vector effectively. Let's go through the steps together.

Step 1: Break Down the Vectors

We have two vectors: vector A with a magnitude of 30 m/s at an angle of 45 degrees, and vector B with a magnitude of 40 m/s at an angle of 80 degrees. We will calculate the horizontal (x) and vertical (y) components for both vectors.

Calculating Components for Vector A

For vector A:

• Horizontal component (Ax) = A * cos(θ) = 30 * cos(45°)
• Vertical component (Ay) = A * sin(θ) = 30 * sin(45°)

Using the values:

• Ax = 30 * (√2/2) ≈ 21.21 m/s
• Ay = 30 * (√2/2) ≈ 21.21 m/s

Calculating Components for Vector B

For vector B:

• Horizontal component (Bx) = B * cos(θ) = 40 * cos(80°)
• Vertical component (By) = B * sin(θ) = 40 * sin(80°)

Using the values:

• Bx = 40 * 0.1736 ≈ 6.94 m/s
• By = 40 * 0.9848 ≈ 39.39 m/s

Step 2: Sum the Components

Now that we have the components for both vectors, we can find the resultant components by adding the corresponding components together:

• Resultant horizontal component (Rx) = Ax + Bx = 21.21 + 6.94 ≈ 28.15 m/s
• Resultant vertical component (Ry) = Ay + By = 21.21 + 39.39 ≈ 60.60 m/s

Step 3: Calculate the Resultant Velocity

To find the magnitude of the resultant vector, we use the Pythagorean theorem:

Resultant velocity (R) = √(Rx² + Ry²)

Substituting the values:

R = √(28.15² + 60.60²) ≈ √(793.62 + 3672.36) ≈ √(4465.98) ≈ 66.78 m/s

Step 4: Determine the Direction of the Resultant Velocity

To find the angle (φ) of the resultant vector with respect to the horizontal, we can use the tangent function:

tan(φ) = Ry / Rx

φ = arctan(Ry / Rx) = arctan(60.60 / 28.15)

Calculating this gives:

φ ≈ 65.57 degrees

Final Result

In summary, the resultant velocity of the two vectors is approximately:

• Magnitude: 66.78 m/s
• Direction: 65.57 degrees above the horizontal

This method of using the parallelogram rule allows us to visualize how two vectors combine to form a resultant vector, giving us both the magnitude and direction of the overall motion. If you have any further questions or need clarification on any step, feel free to ask!