To calculate the total force applied to the car, we need to consider both the force required to accelerate the car and the force required to overcome the incline.
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Convert the final speed to meters per second: \[ 80 , \text{km/h} = \frac{80 \times 1000 , \text{m}}{3600 , \text{s}} = \frac{80000}{3600} \approx 22.22 , \text{m/s} \]
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Calculate the acceleration: The car starts from rest (initial speed \( u = 0 , \text{m/s} \)) and reaches a speed of 22.22 m/s in 90 seconds. \[ a = \frac{v - u}{t} = \frac{22.22 , \text{m/s} - 0 , \text{m/s}}{90 , \text{s}} \approx 0.2469 , \text{m/s}^2 \]
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Calculate the total mass of the car: The mass \( m \) of the car is given as 900 kg.
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Calculate the net force required for acceleration using Newton's second law: \[ F_{acc} = m \cdot a = 900 , \text{kg} \cdot 0.2469 , \text{m/s}^2 \approx 222.21 , \text{N} \]
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Calculate the incline angle: The incline ratio of 1:40 means the rise is 1 meter for every 40 meters of horizontal distance. Using trigonometry: \[ \tan(\theta) = \frac{1}{40} \implies \theta \approx \tan^{-1}(0.025) \implies \theta \approx 1.43^{\circ} \]
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Calculate the gravitational force acting down the incline: The gravitational force down the incline can be calculated as: \[ F_{gravity} = m \cdot g \cdot \sin(\theta) \] where \( g \approx 9.81 , \text{m/s}^2 \) is the acceleration due to gravity, \[ F_{gravity} = 900 , \text{kg} \cdot 9.81 , \text{m/s}^2 \cdot \sin(1.43^{\circ}) \approx 900 \cdot 9.81 \cdot 0.025 \approx 22.1 , \text{N} \]
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Calculate the total force applied: The total force \( F_{total} \) required to both accelerate the car and overcome the gravitational force down the incline is: \[ F_{total} = F_{acc} + F_{gravity} \approx 222.21 , \text{N} + 22.1 , \text{N} \approx 244.31 , \text{N} \]
Thus, the total force applied to the car is approximately 244.31 N.