It takes a force of 1280 N to keep a 1500 kg car moving with constant speed up a slope of
5.000
. If the engine delivers 50.0hp to the drive wheels, what is the maximum speed of the
car?
To find the maximum speed of the car, we can use the following steps:
Step 1: Convert horsepower (hp) to watts (W)
1 horsepower (hp) = 745.7 watts (W)
Therefore, 50.0 hp = 50.0 × 745.7 = 37,285 W
Step 2: Calculate the force of friction opposing the car's motion on the slope
The force of friction opposing the car's motion can be determined using the formula:
Force of friction = mass × acceleration due to gravity × sin(angle of the slope)
Given:
Mass of the car (m) = 1500 kg
Acceleration due to gravity (g) = 9.81 m/s^2
Angle of the slope (θ) = 5.000 degrees
First, convert the angle from degrees to radians:
angle (radians) = angle (degrees) × π / 180
θ = 5.000 × π / 180 = 0.0873 radians
Now, calculate the force of friction:
Force of friction = 1500 kg × 9.81 m/s^2 × sin(0.0873)
Force of friction = 1323.64 N
Step 3: Calculate the net force acting on the car
The net force can be calculated by subtracting the force of friction from the force applied by the engine:
Net force = Force applied - Force of friction = 1280 N - 1323.64 N = -43.64 N (opposite to the direction of motion)
Step 4: Calculate the maximum speed of the car
Using the equation for net force:
Net force = mass × acceleration
Acceleration = Net force / mass
Acceleration = -43.64 N / 1500 kg
Acceleration = -0.02909 m/s^2
Now, we can use the equation for constant speed motion on a slope:
Force applied - Force of friction = mass × acceleration due to gravity × sin(θ)
Rearranging the equation for force applied:
Force applied = Force of friction + mass × acceleration due to gravity × sin(θ)
Force applied = 1323.64 N + 1500 kg × 9.81 m/s^2 × sin(0.0873)
Substituting the values, we get:
Force applied = 1323.64 N + 13348.69 N ≈ 14672.33 N
Now, we can use the equation for work done by the engine:
Work done = Force applied × distance
Work done = 14672.33 N × distance
Since the car is moving with a constant speed, the work done by the engine is equal to the work done to overcome the force of friction:
Work done = Force of friction × distance
14672.33 N × distance = 1323.64 N × distance
Cancelling out distances on both sides of the equation:
14672.33 N = 1323.64 N
Now, we can solve for the distance which will cancel out on both sides:
distance = 1323.64 N / 14672.33 N
Therefore, the maximum speed of the car is approximately 0.0902 times the speed of light.
To find the maximum speed of the car, we can use the concept of work and power.
First, let's convert the power given in horsepower to watts (SI unit).
1 horsepower = 745.7 watts
So, 50.0 horsepower is equal to 50.0 * 745.7 = 37,285 watts.
Now, we can calculate the work done by the engine on the car.
Work = Force * distance
In this case, the distance is not given, but we can find it using the concept of the slope.
The slope is given as 5.000, which means that for every 5.000 meters of horizontal distance, there is a 1.000 meter increase in elevation. In other words, the slope forms a right triangle, where the base is 5.000 meters and the height is 1.000 meter.
Using the Pythagorean theorem, we can find the hypotenuse of the triangle, which represents the distance along the slope.
Hypotenuse = √(base^2 + height^2) = √(5.000^2 + 1.000^2) ≈ 5.099 meters
Now, we can calculate the work done by the engine:
Work = Force * distance = 1280 N * 5.099 m = 6531.52 Joules
We know that power is the rate at which work is done. So, we can calculate the time taken to do the work:
Time = Work / Power = 6531.52 J / 37,285 W ≈ 0.175 seconds
Finally, we can find the maximum speed of the car using the formula:
Speed = Distance / Time
Since the distance is the hypotenuse of the slope, which we found earlier as 5.099 meters, and the time is 0.175 seconds, we can calculate:
Speed = 5.099 m / 0.175 s ≈ 29.140 m/s
Therefore, the maximum speed of the car is approximately 29.140 m/s.