Calculate the thermal energy dissipated from brakes in a 1310-kg car that descends a 13.5° hill. The car begins braking when its speed is 88 km/h and slows down to a speed of 34 km/h in a distance of 0.52 km measured along the road.
thermal energy dissipated= change in KE+change in PE
= 1/2 m(vi^2-vf^2)+mg520*sin13.5
check my thinking.
work done = force * distance
F = m a
Change in potential energy = - m g (.52*10^3) sin 13.5
Change in kinetic energy = - (1/2) m (88-34)^2 (10^3 m/km/3600s/hr)^2
thermal energy + change in potential energy + change in kinetic energy = 0
To calculate the thermal energy dissipated from brakes in a car, we need to find the work done by the brakes and then convert it into thermal energy.
Step 1: Find the gravitational potential energy difference:
The gravitational potential energy difference is given by:
ΔPE = mgh
where m is the mass of the car (1310 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the vertical height of the hill.
To find h, we use trigonometry:
h = d * sin(θ)
where d is the distance along the road (0.52 km) and θ is the angle of the hill (13.5°).
Converting the values to SI units:
m = 1310 kg
g = 9.8 m/s^2
h = (0.52 km) * (1000 m/km) * sin(13.5°)
Step 2: Find the initial and final kinetic energies:
The initial kinetic energy is given by:
KE_initial = (1/2) * m * v^2
where v is the initial speed (88 km/h).
Converting the value to SI units:
v_initial = (88 km/h) * (1000 m/km) * (1 h/3600 s)
The final kinetic energy is given by:
KE_final = (1/2) * m * v^2
where v is the final speed (34 km/h).
Converting the value to SI units:
v_final = (34 km/h) * (1000 m/km) * (1 h/3600 s)
Step 3: Find the work done by the brakes:
The work done by the brakes is given by:
Work = ΔKE + ΔPE
where ΔKE is the change in kinetic energy and ΔPE is the change in potential energy.
ΔKE = KE_final - KE_initial
ΔPE = -ΔPE (since the car is descending)
Step 4: Convert the work done into thermal energy:
The work done by the brakes is converted into thermal energy, assuming no other forms of energy loss.
Thermal Energy = Work
Now, let's calculate all the values and find the thermal energy dissipated from the brakes.
Calculations:
h = (0.52 km) * (1000 m/km) * sin(13.5°)
v_initial = (88 km/h) * (1000 m/km) * (1 h/3600 s)
v_final = (34 km/h) * (1000 m/km) * (1 h/3600 s)
ΔKE = (1/2) * 1310 kg * (v_final^2 - v_initial^2)
ΔPE = -1310 kg * 9.8 m/s^2 * h
Work = ΔKE + ΔPE
Thermal Energy = Work
Just perform the calculations to find the thermal energy dissipated from the brakes.
To calculate the thermal energy dissipated from the brakes, we need to know the work done by the brakes and then convert it into thermal energy.
First, let's calculate the work done by the brakes. Work is defined as the change in potential energy, so we need to determine the change in gravitational potential energy as the car descends the hill.
The change in potential energy can be calculated using the formula:
ΔPE = m * g * Δh
Where:
ΔPE is the change in potential energy
m is the mass of the car (1310 kg)
g is the acceleration due to gravity (9.8 m/s^2)
Δh is the change in height
To calculate Δh, we can use trigonometry. The vertical change in height (Δh) can be determined by multiplying the distance along the road (0.52 km) by sin(13.5°) since the angle is measured between the road and the horizontal line.
Δh = 0.52 km * sin(13.5°)
Next, we need to convert the distance and mass from their given units to SI units:
- Convert 0.52 km to meters: 1 km = 1000 m, so 0.52 km = 0.52 * 1000 m = 520 m
Substituting the values into the equation for Δh:
Δh = 520 m * sin(13.5°)
Now, we can calculate ΔPE:
ΔPE = 1310 kg * 9.8 m/s^2 * 520 m * sin(13.5°)
Once we have the change in potential energy, we can calculate the work done by the brakes using the formula:
Work = -ΔPE
This is because the work done on the car by the brakes is equal to the negative change in potential energy. This negative sign indicates that work is done against the force of gravity.
Finally, to convert the work done by the brakes into thermal energy, we can assume that all the work is converted into heat. Therefore, the thermal energy dissipated by the brakes is equal to the work done.
So, by calculating the work done by the brakes, we can determine the thermal energy dissipated.