a motor car having a mass of 2 tonnes is being accelerated up a gradient of 1 in 25. at an engine speed of 4 000 rimin the b.p. is 57 kw the effective tyre radius is 330 mm, the rolling res?stance is 180 n/tonne, and the rear-axle ratio is 4.4 to 1 assuming a transmission efficiency of 90 % and neglecting air res?stance, determine the value of the acceleration at the given speed 262 mls1
To determine the value of the acceleration at the given speed of 262 m/s, we need to consider the forces acting on the motor car.
First, let’s calculate the force due to the gradient. The gradient of 1 in 25 means that for every 25 units of horizontal distance, there is a rise of 1 unit. Since the motor car is being accelerated up the gradient, the force due to the gradient will act in the opposite direction of motion. The force due to the gradient can be calculated using the formula:
Force due to gradient = mass * gravitational acceleration * sin(theta)
In this case, the mass of the car is 2 tonnes, which is equivalent to 2000 kg. The gravitational acceleration is approximately 9.8 m/s^2, and the angle of the gradient can be calculated using the formula:
theta = arctan(1/25)
Plugging in the values, we can calculate the force due to the gradient.
Next, let’s calculate the force due to rolling resistance. Rolling resistance is the resistance encountered by a rolling object, such as a car, due to the deformation of the tires and the friction between the tires and the road. The force due to rolling resistance can be calculated using the formula:
Force due to rolling resistance = rolling resistance coefficient * mass * gravitational acceleration
In this case, the rolling resistance coefficient is given as 180 N/tonne, which is equivalent to 0.18 N/kg. Plugging in the values, we can calculate the force due to rolling resistance.
Now, let’s calculate the force due to the engine. The power output of the engine is given as 57 kW. Power is the rate at which work is done, and work is force multiplied by distance. The force due to the engine can be calculated using the formula:
Force due to engine = power / velocity
In this case, the velocity is given as 262 m/s. Plugging in the values, we can calculate the force due to the engine.
Finally, let’s calculate the total force acting on the motor car. The total force is the sum of the forces due to the gradient, rolling resistance, and the engine.
Total force = Force due to gradient - Force due to rolling resistance + Force due to engine
Once we have the total force, we can calculate the acceleration using Newton’s second law of motion:
Acceleration = Total force / mass
Plugging in the values, we can calculate the acceleration at the given speed of 262 m/s.