(a) How high a hill can a car coast up (engine disengaged) if work done by friction is negligible
and its initial speed is 110 km/h? (b) If, in actuality, a 750-kg car with an initial speed of 110 km/h
is observed to coast up a hill to a height 22.0 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?
(a) To determine the maximum height a car can coast up a hill with negligible friction, we need to convert the initial speed from km/h to m/s and then calculate the potential energy gained.
1. Convert the initial speed from km/h to m/s:
Given: initial speed = 110 km/h
Conversion: 1 km/h = 0.2778 m/s
Calculation: speed = 110 km/h * 0.2778 m/s = 30.77 m/s
2. Calculate the maximum height using the formula for potential energy:
Formula: Potential Energy (PE) = m * g * h
Given: mass (m) = unknown, acceleration due to gravity (g) = 9.8 m/s^2, height (h) = unknown
Calculation: PE = m * 9.8 * h
However, since the work done by friction is negligible, the initial kinetic energy will be equal to the potential energy gained at the highest point.
Thus, we can equate the kinetic energy at the start to the potential energy at the highest point:
Initial kinetic energy = 0.5 * m * speed^2
Potential energy at the highest point = m * g * h
Equating the two, we get:
0.5 * m * speed^2 = m * g * h
Simplifying and solving for h, we get:
h = (0.5 * speed^2) / g
Plugging in the values, we have:
h = (0.5 * 30.77^2) / 9.8
h ≈ 49.19 meters
Therefore, the car can coast up a hill with a maximum height of approximately 49.19 meters.
(b) To determine the amount of thermal energy generated by friction, we need to calculate the difference between the potential energy gained and the initial kinetic energy.
1. Convert the initial speed from km/h to m/s (same as in part a):
speed = 30.77 m/s
2. Calculate the initial kinetic energy using the formula:
Initial kinetic energy = 0.5 * mass * speed^2
Given: mass = 750 kg
Calculation: Initial kinetic energy = 0.5 * 750 kg * (30.77 m/s)^2
3. Calculate the potential energy gained using the formula:
Potential energy (at highest point) = mass * g * height
Given: height = 22.0 m
Calculation: Potential energy = 750 kg * 9.8 m/s^2 * 22.0 m
4. Calculate the thermal energy generated by friction:
Thermal energy = Potential energy - Initial kinetic energy
Calculation: Thermal energy = (750 kg * 9.8 m/s^2 * 22.0 m) - (0.5 * 750 kg * (30.77 m/s)^2)
Therefore, the thermal energy generated by friction is equal to the calculated value.
(c) To find the average force of friction, we can consider the work done against friction during the car's ascent up the hill.
1. Convert the angle of the slope from degrees to radians:
Given: slope = 2.5°
Conversion: 1° = π/180 radians
Calculation: angle (in radians) = 2.5° * π/180 ≈ 0.0436 rad
2. Calculate the work done against friction:
Work done against friction = Force of friction * distance
Given: slope angle = 0.0436 rad, height = 22.0 m
Calculation: work done = force of friction * distance
Distance = height/sin(slope angle)
Work done = force of friction * (height/sin(slope angle))
The work done against friction is equal to the difference in potential energy:
Work done against friction = Potential energy at highest point - Initial kinetic energy
Calculation: Work done against friction = (750 kg * 9.8 m/s^2 * 22.0 m) - (0.5 * 750 kg * (30.77 m/s)^2)
Rearranging the equation and solving for the force of friction:
Force of friction = (Work done against friction) / Distance
Calculation: Force of friction = [(750 kg * 9.8 m/s^2 * 22.0 m) - (0.5 * 750 kg * (30.77 m/s)^2)] / (22.0 m / sin(0.0436))
Therefore, the average force of friction on the hill slope is equal to the calculated value.
To answer these questions, we need to apply the principles of energy conservation and the concept of work done by friction. I'll guide you through the steps to solve each part of the problem.
(a) To determine how high a hill a car can coast up (engine disengaged), we can use the principle of conservation of mechanical energy. We know that the work done by friction is negligible, so the only energy involved is the car's initial kinetic energy.
1. Convert the initial speed from km/h to m/s. (1 km/h = 1000/3600 m/s)
Initial speed = 110 km/h = (110 * 1000) / 3600 m/s
2. Calculate the initial kinetic energy of the car using the formula:
Kinetic energy = (1/2) * mass * (initial speed)^2
(b) In this part, we need to calculate the thermal energy generated by friction as the car coasts up a hill.
1. Determine the change in potential energy of the car using the formula:
Change in potential energy = mass * gravity * change in height
2. The thermal energy generated by friction is equal to the difference between the initial kinetic energy and the change in potential energy.
(c) To find the average force of friction, we need to consider the forces acting on the car on the inclined slope.
1. Resolve the weight of the car perpendicular and parallel to the slope.
Perpendicular force = weight * cos(slope angle)
Parallel force = weight * sin(slope angle)
2. The average force of friction is equal to the parallel force acting against the motion of the car.
Let's plug in the numbers and calculate the respective results for each part of the problem.