To find the resultant velocity for the total trip, we can break it down into two segments and then find the resultant vector.
-
First Segment:
- Speed: 50 km/h North
- Time: 1 hour
- Distance traveled North = Speed × Time = 50 km/h × 1 h = 50 km
-
Second Segment:
- Speed: 40 km/h West
- Time: 2 hours
- Distance traveled West = Speed × Time = 40 km/h × 2 h = 80 km
Now we have traveled:
- 50 km North
- 80 km West
Resultant Displacement
To determine the resultant displacement, we use the Pythagorean theorem since the paths are perpendicular to each other:
\[ d = \sqrt{(50 , \text{km})^2 + (80 , \text{km})^2} \]
\[ d = \sqrt{2500 + 6400} \]
\[ d = \sqrt{8900} \approx 94.3 , \text{km} \]
Total Time
The total time of the trip is: 1 hour (North) + 2 hours (West) = 3 hours
Average Velocity
Average velocity is given by the formula:
\[ \text{Average velocity} = \frac{\text{Total displacement}}{\text{Total time}} \]
Calculating the average velocity:
\[ \text{Average velocity} = \frac{94.3 , \text{km}}{3 , \text{h}} \approx 31.43 , \text{km/h} \]
Direction of Average Velocity
Now, to find the angle of the resultant vector (northwest), we can use the tangent function:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{50}{80} = 0.625 \]
Calculating the angle:
\[ \theta = \tan^{-1}(0.625) \approx 32^\circ \text{ (North of West)} \]
Conclusion
The resultant average velocity vector for the total trip is approximately 31 km/h heading in a direction northwest.
Among the given answers, the most suitable is 31 km per hour northwest.