A compact car of mass 915 kg has an overall motor efficiency of 16.0%.
(a) If burning one gallon of gasoline supplies 1.34 108 J of energy, find the amount of gasoline used in accelerating the car from rest to 55.0 mi/h. Here you may ignore the effects of air resistance and rolling friction.
(b) How many such accelerations will one gallon provide?
(c) The mileage claimed for the car is 39.0 mi/gal at 55 mi/h. What power is delivered to the wheels (to overcome frictional effects) when the car is driven at this speed?
give answer
To find the amount of gasoline used in accelerating the car, we need to determine the energy required for acceleration.
(a) The energy required for acceleration can be calculated using the equation:
E = (1/2) * m * v^2
Where:
E is the energy required for acceleration
m is the mass of the car
v is the velocity (final speed) of the car
Converting the final speed from mi/h to m/s:
v = 55 mi/h * (1609.34 m/1 mi) * (1 h/3600 s)
v = 24.59 m/s
Plugging in the values:
E = (1/2) * 915 kg * (24.59 m/s)^2
E = 135,107 J
Since the motor efficiency is given as 16.0%, the actual energy provided by the motor to accelerate the car is:
Actual energy = (16.0 / 100) * 135,107 J
Actual energy = 21,617 J
To find the amount of gasoline used, we need to divide the actual energy by the energy provided by burning one gallon of gasoline:
Amount of gasoline used = Actual energy / Energy provided by one gallon of gasoline
Energy provided by one gallon of gasoline = 1.34 * 10^8 J
Amount of gasoline used = 21,617 J / (1.34 * 10^8 J)
Calculating the value:
Amount of gasoline used = 0.000161 gallons
Therefore, the amount of gasoline used in accelerating the car from rest to 55.0 mi/h is approximately 0.000161 gallons.
(b) To find how many such accelerations will one gallon provide, we need to divide the total energy provided by burning one gallon of gasoline by the energy required for acceleration:
Number of accelerations = Energy provided by one gallon of gasoline / Energy required for acceleration
Number of accelerations = 1.34 * 10^8 J / 21,617 J
Calculating the value:
Number of accelerations = 6204
Therefore, one gallon of gasoline will provide approximately 6204 accelerations.
(c) The power delivered to the wheels can be calculated using the formula:
Power = Work / Time
Given that the mileage claimed for the car is 39.0 mi/gal at 55 mi/h, we can calculate the time taken to cover one mile:
Time taken to cover one mile = (1 mile / 55 mi/h) * (3600 s / 1 h)
Time taken to cover one mile = 65.45 s
To overcome frictional effects, the work done is equal to the energy provided by burning one gallon of gasoline, which is 1.34 * 10^8 J.
Plugging in the values:
Power = (1.34 * 10^8 J) / (39.0 mi/gal * 65.45 s/mi)
Calculating the power:
Power = 538.4 W
Therefore, the power delivered to the wheels (to overcome frictional effects) when the car is driven at a speed of 55 mi/h is approximately 538.4 W.
To solve these problems, we need to apply the concept of work and energy.
(a) First, let's find the total energy required to accelerate the car from rest to 55.0 mi/h.
1. Convert the initial velocity from miles per hour (mi/h) to meters per second (m/s).
1 mi/h = 0.44704 m/s
55.0 mi/h = 24.5872 m/s
2. Calculate the kinetic energy (KE) of the car at 55.0 mi/h.
KE = (1/2) * mass * velocity^2
KE = (1/2) * 915 kg * (24.5872 m/s)^2
3. Calculate the work done on the car using the efficiency of the motor.
Efficiency = (useful work output) / (total work input)
Since we are looking for the total work input, we rearrange the equation:
Total work input = (useful work output) / Efficiency
The useful work output is equal to the change in kinetic energy:
Useful work output = ΔKE = KE_final - KE_initial
Total work input = (ΔKE) / Efficiency
(b) To find how many such accelerations one gallon of gasoline can provide, we need to calculate the energy provided by burning one gallon of gasoline.
1. Given energy from burning one gallon of gasoline = 1.34 * 10^8 J
2. Divide the total work input from part (a) by the energy from burning one gallon of gasoline to find the number of accelerations.
Number of accelerations = (Total work input) / (Energy from burning one gallon of gasoline)
(c) To find the power delivered to overcome frictional effects at a speed of 55 mi/h, we can use the formula for power.
Power = Force * Velocity
In this case, the force is the frictional force, which we can find using the formula: Frictional force = Mass * Acceleration
1. Convert the speed from miles per hour (mi/h) to meters per second (m/s).
2. Calculate the acceleration of the car assuming it has reached a constant speed.
Since the car is moving at a constant speed, the net force acting on it is zero. Therefore, the frictional force is equal to the force produced by the engine.
Frictional force = Force = Mass * Acceleration
Solving for acceleration: Acceleration = Force / Mass
3. Calculate the power delivered to the wheels using the formula Power = Force * Velocity.
Now let's plug in the values and calculate the answers.