Proper design of automobile braking systems must account for heat buildup under heaving braking. Calculate the thermal energy dissipated from brakes in a 1460-kg car that descends a 15.5° hill. The car begins braking when its speed is 92 km/h and slows down to a speed of 28 km/h in a distance of 0.44 km measured along the road.
To calculate the thermal energy dissipated from the brakes, we can follow these steps:
Step 1: Calculate the change in kinetic energy of the car:
Using the formula for kinetic energy, we have:
Initial kinetic energy = (1/2) * mass * initial velocity^2
Final kinetic energy = (1/2) * mass * final velocity^2
Initial velocity = 92 km/h = 92 * 1000 m / (3600 s) = 25.56 m/s
Final velocity = 28 km/h = 28 * 1000 m / (3600 s) = 7.78 m/s
Initial kinetic energy = (1/2) * 1460 kg * (25.56 m/s)^2
Final kinetic energy = (1/2) * 1460 kg * (7.78 m/s)^2
Step 2: Calculate the change in potential energy of the car:
Using the formula for potential energy, we have:
Initial potential energy = mass * g * initial height
Final potential energy = mass * g * final height
Mass = 1460 kg
g = acceleration due to gravity = 9.8 m/s^2
Initial height = 0 m (as the car starts at the bottom of the hill)
Final height = distance * sin(theta)
Distance = 0.44 km = 0.44 * 1000 m = 440 m
theta = 15.5° = 15.5 * (pi / 180) rad
Final height = 440 m * sin(15.5°)
Step 3: Calculate the total energy dissipated as heat:
Total energy dissipated = change in kinetic energy + change in potential energy
Total energy dissipated = (Initial kinetic energy - Final kinetic energy) + (Final potential energy - Initial potential energy)
Finally, substitute the values from the previous steps and calculate the thermal energy dissipated from the brakes.
To calculate the thermal energy dissipated from the brakes, we need to first calculate the change in kinetic energy of the car as it slows down and then convert this into thermal energy.
1. Find the change in kinetic energy (ΔKE):
- Convert the initial and final speeds from km/h to m/s:
Initial speed (v₁) = 92 km/h = 92 × (1000/3600) m/s = 25.56 m/s
Final speed (v₂) = 28 km/h = 28 × (1000/3600) m/s = 7.78 m/s
- Calculate the change in kinetic energy using the formula:
ΔKE = (1/2) × mass × ((v₂)² - (v₁)²)
2. Convert the change in kinetic energy into thermal energy:
- Convert the mass of the car from kg to g (grams):
mass = 1460 kg = 1460 × 1000 g = 1,460,000 g
- Convert ΔKE from J (joules) to cal (calories):
1 J = 0.239005736 cal
ΔKE (cal) = ΔKE (J) / 0.239005736
3. Calculate the thermal energy dissipated:
- The thermal energy dissipated (Q) is equal to the change in kinetic energy:
Q = ΔKE (cal)
That's it! Now let's calculate the values:
1. Change in kinetic energy (ΔKE):
ΔKE = (1/2) × 1,460,000 g × ((7.78 m/s)² - (25.56 m/s)²)
2. Convert ΔKE from J to cal:
ΔKE (cal) = ΔKE (J) / 0.239005736
3. Thermal energy dissipated (Q):
Q = ΔKE (cal)
By following these steps, you can calculate the thermal energy dissipated from the brakes of the car.