To solve this problem, let's break it down step by step:
### Step 1: Analyze the Forces Acting on the Car
When the car is on an incline, it experiences two main forces:
1. The applied force parallel to the incline ([tex]\( F = 8200 \)[/tex] N)
2. The gravitational force, which can be decomposed into two components:
- The component perpendicular to the incline (which does not affect the car's acceleration along the incline)
- The component parallel to the incline
The component of the gravitational force parallel to the incline can be calculated using the formula:
[tex]\[ F_{\text{gravity parallel}} = mg \sin(\theta) \][/tex]
where:
- [tex]\( m = 1800 \)[/tex] kg (mass of the car)
- [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex] (acceleration due to gravity)
- [tex]\( \theta = 30^{\circ} \)[/tex] (incline angle)
### Step 2: Calculate the Gravitational Force Parallel to the Incline
Using the formula:
[tex]\[ F_{\text{gravity parallel}} = 1800 \times 9.8 \times \sin(30^{\circ}) \][/tex]
Since [tex]\(\sin(30^{\circ}) = 0.5\)[/tex]:
[tex]\[ F_{\text{gravity parallel}} = 1800 \times 9.8 \times 0.5 = 1800 \times 4.9 = 8819.999999999998 \ \text{N} \][/tex]
### Step 3: Determine the Net Force Parallel to the Incline
The net force acting on the car along the incline is the difference between the applied force and the parallel component of the gravitational force:
[tex]\[ F_{\text{net parallel}} = 8200 \, \text{N} - 8819.999999999998 \, \text{N} = -619.9999999999982 \, \text{N} \][/tex]
The negative sign indicates that the net force is acting in the opposite direction to the applied force.
### Step 4: Calculate the Acceleration
Using Newton's second law ([tex]\( F = ma \)[/tex]):
[tex]\[ a = \frac{F_{\text{net parallel}}}{m} = \frac{-619.9999999999982}{1800} = -0.34444444444444344 \, \text{m/s}^2 \][/tex]
### Conclusion
The car's new acceleration on the incline is [tex]\(-0.344 \, \text{m/s}^2\)[/tex], which means it is decelerating due to the gravitational pull.
The closest provided option is:
C. [tex]\( -2.5 \, \text{m/s}^2 \)[/tex].
However, since none of the given options perfectly match the derived acceleration, it indicates there might be an error or discrepancy in the list of provided options. The correct acceleration, based on the calculations, is approximately [tex]\(-0.34 \, \text{m/s}^2\)[/tex].
For the provided options, none is accurate, but the nearest considered would be option C if one must be chosen.
