(a) How high a hill can a car coast up (engine disengaged) if friction is negligible and its initial speed is 96.0 km/h?
(b) If, in actuality, a 750 kg car with an initial speed of 96.0 km/h is observed to coast up a hill to a height 18.0 m above its starting point, how much thermal energy was generated by friction?
(c) What is the average force of friction if the hill has a slope 2.5° above the horizontal? (Explicitly show on paper how you follow the steps in the Problem-Solving Strategy for energy found on pages 159 and 160. Your instructor may ask you to turn in this work.)
N (down the slope)
(a) To determine how high a hill a car can coast up with the engine disengaged, we can use the principle of conservation of mechanical energy. The initial mechanical energy of the car is equal to its final mechanical energy when it reaches the highest point of the hill. Friction is neglected in this scenario.
The initial mechanical energy (Ei) of the car is given by:
Ei = (1/2) * m * v^2
where m represents the mass of the car and v is its initial speed.
In this case, the mass of the car is not given, but we can assume a value for the mass (e.g., 1000 kg) for the calculation. The initial speed is given as 96.0 km/h, which needs to be converted to m/s:
v = 96.0 km/h * (1000 m/km) / (3600 s/h) ≈ 26.7 m/s
Substituting the values into the equation, we have:
Ei = (1/2) * 1000 kg * (26.7 m/s)^2 = 357,450 J
At the highest point of the hill, the final mechanical energy (Ef) is given by:
Ef = m * g * h
where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the height of the hill.
Since we want to find the maximum height the car can coast up, Ef will be equal to Ei. Solving for h:
h = Ei / (m * g)
Using the assumed mass of 1000 kg, we have:
h = 357,450 J / (1000 kg * 9.8 m/s^2) ≈ 36.5 m
Therefore, the car can coast up a hill with a maximum height of approximately 36.5 meters.
(b) To calculate the amount of thermal energy generated by friction when the car actually coasted up a hill to a height of 18.0 m, we need to determine the change in mechanical energy (ΔE) of the car. This change is due to the thermal energy generated by friction.
Using the equation for ΔE:
ΔE = Ef - Ei
Initial mechanical energy (Ei) is the same as in part (a), which is 357,450 J. The final mechanical energy (Ef) is given by:
Ef = m * g * h
where h is the height of the hill, given as 18.0 m.
Again, using the assumed mass of 1000 kg:
Ef = 1000 kg * 9.8 m/s^2 * 18.0 m = 176,400 J
Substituting the values into the equation for ΔE:
ΔE = 176,400 J - 357,450 J ≈ -181,050 J
The negative sign indicates that the system (car and hill) lost mechanical energy, which is converted into thermal energy due to friction.
Therefore, the amount of thermal energy generated by friction is approximately 181,050 J.
(c) To determine the average force of friction on the car while coasting up the hill with a slope of 2.5°, we need to consider the work done by friction. We can use the equation:
W = F * d * cos(θ)
where W is the work done, F is the force of friction, d is the displacement of the car up the hill, and θ is the angle between the direction of the force and the displacement.
In this case, the displacement of the car up the hill is equal to the height of the hill, which is 18.0 m. The angle θ is given as 2.5°.
Converting the angle to radians:
θ = 2.5° * (π/180°) ≈ 0.044 radians
The work done by friction is equal to the change in mechanical energy, which we calculated in part (b) as approximately -181,050 J.
Substituting the values into the equation for work done:
-181,050 J = F * 18.0 m * cos(0.044 radians)
Solving for the force of friction:
F = -181,050 J / (18.0 m * cos(0.044 radians))
Calculating the force of friction using a calculator, we find:
F ≈ -1717 N (rounded to the nearest whole number)
Therefore, the average force of friction on the car while coasting up the hill is approximately 1717 Newtons in the downward direction.
(a) To determine how high a hill a car can coast up with negligible friction, we can use the conservation of mechanical energy principle. The initial kinetic energy of the car will be converted into potential energy as it moves up the hill.
The equation for conservation of mechanical energy is given as:
KE_initial + PE_initial = KE_final + PE_final
Since the car starts with an initial speed of 96.0 km/h, we need to convert it to meters per second.
1 km/h = 1000 m / 3600 s = 1/3.6 m/s
Therefore, the initial speed of the car is:
96.0 km/h * 1/3.6 m/s = 26.67 m/s
The initial potential energy of the car is zero since it starts at the bottom of the hill, so the equation becomes:
KE_initial = KE_final + PE_final
1/2 * m * v^2 = m * g * h
Here, m is the mass of the car, v is the final velocity (which is zero as the car comes to rest), g is the acceleration due to gravity, and h is the height of the hill.
Rearranging the equation to solve for h:
h = (v^2) / (2g)
Plugging in the values:
h = (26.67 m/s)^2 / (2 * 9.8 m/s^2) = 35.247 m
Therefore, the car can coast up a hill with a maximum height of 35.247 meters.
(b) To determine the thermal energy generated by friction, we need to calculate the work done by friction as the car moves up the hill. The work done by friction is given as:
Work_friction = Force_friction * distance
The force of friction can be determined using the equation:
Force_friction = mass * acceleration
Since the car is observed to coast up a hill to a height of 18.0 m, the work done by friction is equal to the change in potential energy:
Work_friction = m * g * h
The thermal energy generated by friction is equal to the work done by friction. Therefore:
Thermal energy = m * g * h = 750 kg * 9.8 m/s^2 * 18.0 m = 132300 J
Therefore, the thermal energy generated by friction is 132300 Joules.
(c) To determine the average force of friction, we need to consider the work done by friction and the distance over which this work is done. The work done by friction is given as:
Work_friction = Force_friction * distance
Since the force of friction is acting in the direction opposite to the displacement, the work done by friction will be negative. Therefore:
Work_friction = -m * g * d
Here, d is the distance over which the force of friction acts. In this case, it is the distance along the slope of the hill.
The average force of friction can be determined by rearranging the equation:
Force_friction = Work_friction / distance
Now, let's calculate the distance along the slope:
distance = (height) / (sin θ)
Here, θ is the angle of the slope, which is given as 2.5°. Converting it to radians:
θ = 2.5° * (π/180°) = 0.04363 rad
Plugging in the values:
distance = 18.0 m / sin(0.04363) = 391.8 m
Now, let's calculate the average force of friction:
Force_friction = (-750 kg * 9.8 m/s^2 * 391.8 m) / 391.8 m = -7350 N
Note that the force of friction is negative since it opposes the motion of the car up the hill. Therefore, the average force of friction is approximately 7350 N downwards along the slope.