To solve the problem of finding the change in velocity of the car moving east and then north, let's break down the steps in detail:
### Step-by-Step Solution:
1. Determine the Initial Velocities:
- The car moves with a constant speed of 40 m/s towards the east.
- The car then moves with a constant speed of 30 m/s towards the north.
2. Resultant Speed Calculation:
- To find the net change in velocity (resultant velocity), we can treat the eastward and northward speeds as the legs of a right triangle. We then use the Pythagorean theorem to determine the magnitude of the resultant velocity.
- [tex]\( \text{Resultant Speed} = \sqrt{(40\, \text{m/s})^2 + (30\, \text{m/s})^2} \)[/tex]
3. Substituting the given values:
- [tex]\( \text{Resultant Speed} = \sqrt{1600 + 900} \)[/tex]
- [tex]\( \text{Resultant Speed} = \sqrt{2500} \)[/tex]
- [tex]\( \text{Resultant Speed} = 50\, \text{m/s} \)[/tex]
4. Resultant Direction Calculation:
- To find the direction of the resultant velocity, we use trigonometry. Specifically, we use the arctangent function to determine the angle relative to the east direction.
- [tex]\( \theta = \tan^{-1} \left( \frac{\text{Speed North}}{\text{Speed East}} \right) \)[/tex]
- [tex]\( \theta = \tan^{-1} \left( \frac{30}{40} \right) \)[/tex]
5. Calculating the angle:
- [tex]\( \theta = \tan^{-1} (0.75) \)[/tex]
- [tex]\( \theta \approx 36.87^\circ \)[/tex]
6. Directional Adjustment:
- The angle of [tex]\( 36.87^\circ \)[/tex] is measured from the east direction towards the north.
- To determine the direction with respect to the reference options provided, we need to convert this angle to a standard bearing which is typically measured clockwise from the north.
- Since [tex]\( 36.87^\circ \)[/tex] is from the east towards north, it implies that measured counterclockwise from north, it would be [tex]\( 90^\circ - 36.87^\circ \)[/tex]
7. Converting to Bearing:
- [tex]\( 90^\circ - 36.87^\circ = 53.13^\circ \)[/tex] (north of east)
- To convert this to a bearing which we measure clockwise from north:
- [tex]\( 360^\circ - 53.13^\circ = 306.87^\circ \)[/tex]
### Conclusion:
The resultant change in velocity of the car is:
Magnitude: 50 m/s
Direction: 307° (approximately)
Thus, the correct answer is:
D 50 m/s at 307°
