Questions: An airplane starts moving with a thrust force of 434 Newtons at an angle 14 degrees. It experiences air resistance of 23 Newtons at an angle 149 degrees. Find the MAGNITUDE of the resultant force acting on the plane. Give your answer in Newtons, however do not explicitly include any units in your answer (that is, do not include an " N " in your answer, enter a numerical answer only).

Solution Steps

The thrust force is given as 434 N at an angle of 14 degrees. We resolve this force into its horizontal and vertical components using trigonometric functions:

• Horizontal component: \( F_{T_x} = 434 \cos(14^\circ) \)
• Vertical component: \( F_{T_y} = 434 \sin(14^\circ) \)

The air resistance is given as 23 N at an angle of 149 degrees. We resolve this force into its horizontal and vertical components:

• Horizontal component: \( F_{R_x} = 23 \cos(149^\circ) \)
• Vertical component: \( F_{R_y} = 23 \sin(149^\circ) \)

The resultant force components are found by summing the respective components of the thrust and air resistance forces:

• Resultant horizontal component: \( F_{x} = F_{T_x} + F_{R_x} \)
• Resultant vertical component: \( F_{y} = F_{T_y} + F_{R_y} \)

The magnitude of the resultant force is calculated using the Pythagorean theorem:

\[ F_{\text{resultant}} = \sqrt{F_{x}^2 + F_{y}^2} \]

1. Calculate the components of the thrust force:

   • \( F_{T_x} = 434 \cos(14^\circ) \approx 421.2765 \)
   • \( F_{T_y} = 434 \sin(14^\circ) \approx 105.0635 \)

2. Calculate the components of the air resistance:

   • \( F_{R_x} = 23 \cos(149^\circ) \approx -19.7985 \)
   • \( F_{R_y} = 23 \sin(149^\circ) \approx 12.0645 \)

3. Calculate the resultant components:

   • \( F_{x} = 421.2765 - 19.7985 = 401.4780 \)
   • \( F_{y} = 105.0635 + 12.0645 = 117.1280 \)

4. Calculate the magnitude of the resultant force: \[ F_{\text{resultant}} = \sqrt{401.4780^2 + 117.1280^2} \approx 418.1561 \]

Final Answer