The driver of a 1,450kg car, whose iron brake calipers mass is 3kg endures a brain malfuction at the top of a 20 degree parking ramp 20 meters long. It approaches the ramp going 20m/s and the driver recovers just as the car reaches the bottom. If the driver captures all the energy of this catastrophe in his car's calipers, then how much hotter do they become.
To determine how much hotter the car's iron brake calipers become, we can calculate the change in thermal energy using the principle of conservation of energy.
1. Calculate the change in gravitational potential energy (GPE) of the car as it moves down the ramp.
GPE = mass * acceleration due to gravity * change in height
The change in height can be calculated using trigonometry, as the ramp is at a 20-degree angle.
Change in height = 20 meters * sin(20 degrees)
2. Calculate the kinetic energy (KE) of the car just before it reaches the bottom of the ramp.
KE = (1/2) * mass * velocity^2
3. The change in thermal energy (ΔE) can be equated to the change in GPE and KE, as the driver captures all the energy of the catastrophe.
ΔE = ΔGPE + ΔKE
4. Calculate the change in temperature (ΔT) using the specific heat capacity of iron.
ΔT = ΔE / (mass of iron calipers * specific heat capacity of iron)
Let's calculate the values step by step:
Mass of the car (m) = 1,450 kg
Mass of the iron brake calipers (m_calipers) = 3 kg
Acceleration due to gravity (g) = 9.8 m/s^2
Length of the ramp (l) = 20 m
Angle of the ramp (θ) = 20 degrees
Velocity of the car (v) = 20 m/s
Specific heat capacity of iron (c) = typically around 450 J/kg°C
1. Change in height (Δh):
Δh = l * sin(θ)
2. Change in GPE (ΔGPE):
ΔGPE = m * g * Δh
3. Kinetic energy (KE):
KE = (1/2) * m * v^2
4. Change in thermal energy (ΔE):
ΔE = ΔGPE + ΔKE
5. Change in temperature (ΔT):
ΔT = ΔE / (m_calipers * c)
By plugging in the given values into each equation, you can calculate the change in temperature (ΔT) of the iron brake calipers.