A sports car with a very powerful engine driving on a surface with a coefficient of static
friction µs = 0.8 and a ten degree incline wants to accelerate from a stop to 100 km/hr. (You may neglect air resistance.)
a) What is the fastest that the car can accomplish this?
b) How far will the car travel in this time?
c) Suppose that the mass of the car is 1500 kg. What is the maximum power output required
from the engine in order to accomplish this?
To solve this problem, we need to use the Newton's Second Law of Motion and the principles of friction and work.
Let's begin step-by-step:
a) To find the fastest acceleration the car can achieve, we need to determine the maximum force of friction that can be applied to the car without it slipping on the incline.
The formula for the maximum force of friction (Fmax) is:
Fmax = µs * N
Where µs is the coefficient of static friction and N is the normal force.
The normal force (N) can be calculated using the equation:
N = mg * cos(θ)
Where m is the mass of the car, g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of the incline (10 degrees).
Substituting the given values, we have:
m = 1500 kg
g = 9.8 m/s²
θ = 10 degrees
N = 1500 kg * 9.8 m/s² * cos(10 degrees)
N = 14697.06 N
Now, let's calculate Fmax:
Fmax = 0.8 * 14697.06 N
Fmax ≈ 11757.65 N
Next, we can use Newton's second law of motion to determine the car's maximum possible acceleration (a):
Fmax = ma
a = Fmax / m
a = 11757.65 N / 1500 kg
a ≈ 7.85 m/s²
Now that we have the acceleration, we can find the time it takes for the car to reach its maximum speed using the following equation:
v = u + at
Where v is the final velocity, u is the initial velocity (assumed to be 0 m/s as the car is starting from a stop), a is the acceleration, and t is the time.
We want v to be 100 km/hour (which is 27.78 m/s):
27.78 m/s = 0 + (7.85 m/s²) * t
Now, we can solve for t:
t = 27.78 m/s / 7.85 m/s²
t ≈ 3.54 seconds
So, the fastest time the car can accomplish accelerating from 0 to 100 km/hr is approximately 3.54 seconds.
b) To find the distance the car will travel in this time, we can use the equation:
s = ut + (1/2)at²
Where s is the distance, u is the initial velocity, a is the acceleration, and t is the time.
Plugging in the values:
s = 0 * 3.54 + (1/2) * 7.85 m/s² * (3.54 seconds)²
s ≈ 49.5 meters
Therefore, the car will travel approximately 49.5 meters in this time.
c) Finally, to find the maximum power output required from the engine, we can use the equation:
P = F * v
Where P is the power, F is the force, and v is the velocity.
The force required to achieve the car's maximum acceleration is given by:
F = ma
F = 1500 kg * 7.85 m/s²
F ≈ 11775 N
Plugging in the values, we have:
P = 11775 N * 27.78 m/s
P ≈ 327,276.5 Watts or 327.28 kilowatts
Therefore, the maximum power output required from the engine is approximately 327.28 kW.
To answer these questions, we need to use the concepts of friction, acceleration, and power. Here's how you can approach each question:
a) To determine the fastest time for the car to accelerate, we need to find the maximum acceleration it can achieve without breaking the static friction. The formula for maximum acceleration is given by a = µg, where µ is the coefficient of static friction and g is the acceleration due to gravity (approximately 9.8 m/s²).
First, let's convert the velocity from km/hr to m/s:
100 km/hr = (100,000 m) / (3600 s) = 27.78 m/s
Next, calculate the maximum acceleration the car can achieve:
a = µs * g = 0.8 * 9.8 m/s² = 7.84 m/s²
Now, we can find the time it takes for the car to reach this maximum velocity using the formula v = u + at, where v is the final velocity, u is the initial velocity (which is 0 m/s for a stopped car), a is the acceleration, and t is the time:
27.78 m/s = 0 m/s + (7.84 m/s²) * t
Solving for t, we get:
t = 27.78 m/s / 7.84 m/s² = 3.54 seconds
Therefore, the fastest time the car can achieve is approximately 3.54 seconds.
b) To calculate the distance traveled by the car during this time, we can use the formula d = ut + (1/2)at². Since the initial velocity is 0 m/s, the equation simplifies to d = (1/2)at².
Substituting the known values, we get:
d = (1/2) * (7.84 m/s²) * (3.54 s)²
d = 1/2 * 7.84 m/s² * (12.5 s)²
d = 1/2 * 7.84 m/s² * 156.25 s²
d = 61.25 m
Therefore, the car will travel approximately 61.25 meters during this time.
c) To calculate the maximum power output required from the engine, we can use the formula P = Fnet * v, where P is power, Fnet is the net force acting on the car, and v is the velocity.
The net force acting on the car can be calculated using the formula Fnet = m * a, where m is the mass of the car and a is the acceleration.
Substituting the known values, we get:
Fnet = (1500 kg) * (7.84 m/s²) = 11760 N
Now, we can calculate the power:
P = Fnet * v = 11760 N * 27.78 m/s = 326,572.8 Watts
Therefore, the maximum power output required from the engine in order to accomplish this is approximately 326.6 kW.