Proper design of automobile braking systems must account for heat buildup under heavy breaking. Calculate the thermal energy dissipated from brakes in a 1500 kg car that descends a 17° hill. The car begins braking when its speed is 95 km/h and slows to a speed of 35 km/h in a distance of 0.30 km measured along the road.
To calculate the thermal energy dissipated from the brakes, we need to first calculate the work done by the braking force on the car, using the work-energy principle:
The work done by the braking force can be calculated as:
Work (W) = Change in kinetic energy
Now, let's calculate the change in kinetic energy of the car as it slows down from 95 km/h to 35 km/h.
Initial velocity (v1) = 95 km/h = 26.4 m/s
Final velocity (v2) = 35 km/h = 9.7 m/s
Change in kinetic energy = 1/2 * mass * (v2^2 - v1^2)
Mass of the car (m) = 1500 kg
Change in kinetic energy = 1/2 * 1500 kg * (9.7 m/s)^2 - (26.4 m/s)^2
Now, let's calculate the distance covered by the car during braking.
Distance covered (d) = 0.30 km = 300 m
Now, let's calculate the braking force (F) using Newton's second law:
Force (F) = mass * acceleration
The car is descending a 17° hill, which means it experiences an additional force due to gravity. The gravitational force (mg) acting on the car can be calculated as:
Gravitational force (mg) = mass * gravitational acceleration
Gravitational acceleration (g) = 9.8 m/s^2
Gravitational force (mg) = 1500 kg * 9.8 m/s^2
Total force (F) = Force due to gravity (mg) + Braking force
Now, let's calculate the total force (F) using the given information.
Total force (F) = Mass * acceleration
Acceleration = Change in velocity / Time taken
The time taken to reduce the speed from 95 km/h to 35 km/h can be calculated using the average speed formula:
Average speed = Total distance / Total time
The average speed can be calculated as:
Average speed = (Initial velocity + Final velocity) / 2
Let's calculate the total time taken.
Average speed = (v1 + v2) / 2
Total time taken = Total distance / Average speed
Now, let's substitute the known values and calculate the total time taken.
Total time taken = 300 m / ((26.4 m/s + 9.7 m/s) / 2)
Finally, let's calculate the total force (F) using the known values.
Total force (F) = 1500 kg * ((9.7 m/s - 26.4 m/s) / (Total time taken))
Now that you have calculated the total force (F), you can multiply it by the distance covered (d) to calculate the work done by the braking force. Since thermal energy dissipated is equal to the work done, you can determine the thermal energy dissipated from the brakes.