To solve for the resultant velocity of the given vectors, we need to break each vector into its horizontal ($x$-component) and vertical ($y$-component). Then, we can sum all the components to get the resultant vector in both directions, and finally calculate the magnitude and direction of the resultant vector.

Step 1: Break each vector into components

• Vector A:

  • Magnitude: 3.7 m/s, Angle: 0°
  • $A_x = 3.7 \cos(0°) = 3.7$
  • $A_y = 3.7 \sin(0°) = 0$

• Vector B:

  • Magnitude: 1.5 m/s, Angle: 38°
  • $B_x = 1.5 \cos(38°)$
  • $B_y = 1.5 \sin(38°)$

• Vector C:

  • Magnitude: 6.1 m/s, Angle: 90°
  • $C_x = 6.1 \cos(90°) = 0$
  • $C_y = 6.1 \sin(90°) = 6.1$

• Vector D:

  • Magnitude: 2.2 m/s, Angle: 160°
  • $D_x = 2.2 \cos(160°)$
  • $D_y = 2.2 \sin(160°)$

Now, I'll calculate the exact values for $B_x$, $B_y$, $D_x$, and $D_y$.

Step 2: Calculate the components

Let me calculate those.The components of each vector are as follows:

• Vector A:

  • $A_x = 3.7$
  • $A_y = 0.0$

• Vector B:

  • $B_x = 1.18$
  • $B_y = 0.92$

• Vector C:

  • $C_x = 0.0$
  • $C_y = 6.1$

• Vector D:

  • $D_x = -2.07$
  • $D_y = 0.75$

Step 3: Sum the components

To get the resultant vector's components, we add the individual components:

$R_x = A_x + B_x + C_x + D_x$ $R_y = A_y + B_y + C_y + D_y$

Now, let's compute the resultant components.The resultant components are:

• $R_x = 2.81$
• $R_y = 7.78$

Step 4: Calculate the magnitude and direction of the resultant vector

• The magnitude $R$ is given by: $R = \sqrt{R_x^2 + R_y^2}$

• The direction $\theta$ (relative to the positive $x$-axis) is: $\theta = \tan^{-1} \left( \frac{R_y}{R_x} \right)$

Let's compute the magnitude and direction.The resultant velocity is:

• Magnitude: 8.27 m/s
• Direction: 70.10° relative to the positive $x$-axis.

Would you like more details or have any questions?

Related questions:

1. How do you break a vector into components?
2. Why is the angle measured from the positive $x$-axis?
3. What are some common applications of vector addition in physics?
4. How can you add more than two vectors graphically?
5. What is the difference between velocity and speed?

Tip:

Always draw a diagram when working with vectors to visualize the direction and components clearly.