As your vacation job, you are driving a concrete mixer (mass 3300 kg). While driving down a hill (angle α = 29° with the horizontal), suddenly, a cow steps out on the road. You hit on the brakes, and with each locked wheel leaving 24 meters of skid marks, you come to a stop 47 cm before hitting the oblivious cow. The farmer yells at you and claims you have been speeding. Only a quick calculation, based on the length of the skid marks, would convience him that you have been driving responsibly.
You happen to know that the coefficient of kinetic friction between your tires and the road is 1.03. Thus, what was your speed?
Well, first of all, I must say that I find it utterly amusing that a cow decided to take a stroll in front of a concrete mixer. Moo-ve out of the way, cow! Now, let's calculate your speed and clear your name.
Given the information provided, we can start by finding the acceleration due to friction. The equation for the force of friction is:
Frictional Force = coefficient of friction × Normal force
The normal force can be calculated using the equation:
Normal force = mass × gravitational acceleration × cos(α)
Using the values given, we can find the normal force acting on the concrete mixer:
Normal force = 3300 kg × 9.8 m/s² × cos(29°)
Now that we have the normal force, we can calculate the force of friction:
Frictional Force = 1.03 × (3300 kg × 9.8 m/s² × cos(29°))
Once we have the force of friction, we can use Newton's second law of motion:
Force = mass × acceleration
To find the acceleration caused by the force of friction, we rearrange the equation:
acceleration = Frictional Force / mass
With the acceleration known, we can use the kinematic equation to find your initial speed:
vf² = vi² + 2 × acceleration × distance
Here, vf is the final velocity (0 m/s since you stopped), vi is the initial velocity (what we're looking for), acceleration is the negative acceleration from friction, and the distance is the length of the skid marks.
Rearranging the equation, we have:
vi² = -2 × acceleration × distance
Now we can plug in the values:
vi² = -2 × (Frictional Force / mass) × distance
And taking the square root of both sides will give us your initial speed.
So, grab your calculator and let's get calculating! I'm sure once the farmer sees the numbers, he'll be utterly convinced that you were not speeding.
To find the speed at which you were driving, we can use the equation for the skid distance:
skid distance = (initial velocity^2 / (2 * acceleration))
In this case, the skid distance is given as 24 meters and the acceleration can be calculated using the coefficient of kinetic friction and the angle of the hill.
First, let's calculate the acceleration:
acceleration = g * sin(α) - μ * g * cos(α)
where g is the acceleration due to gravity (9.8 m/s^2), α is the angle of the hill (29°), and μ is the coefficient of kinetic friction (1.03).
Let's plug in the values:
acceleration = (9.8 m/s^2) * sin(29°) - (1.03) * (9.8 m/s^2) * cos(29°)
Next, we can use the skid distance and acceleration to find the initial velocity:
24 meters = (initial velocity^2 / (2 * acceleration))
We can rearrange the equation to solve for the initial velocity:
initial velocity^2 = 2 * acceleration * skid distance
initial velocity = sqrt(2 * acceleration * skid distance)
Now, let's plug in the values and calculate the initial velocity:
initial velocity = sqrt(2 * (acceleration) * (skid distance))
Calculating the value:
initial velocity = sqrt(2 * (acceleration) * (skid distance))
initial velocity = sqrt(2 * ((9.8 m/s^2) * sin(29°) - (1.03) * (9.8 m/s^2) * cos(29°)) * (24 meters))
approx. initial velocity ≈ 37.5 m/s
Therefore, your speed was approximately 37.5 m/s when you hit the brakes.
To calculate the initial speed of the concrete mixer, we can use the principles of conservation of energy. Here's how you can calculate it step-by-step.
Step 1: Determine the gravitational potential energy at the top of the hill.
The gravitational potential energy is given by the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the vertical height. In this case, the vertical height is the product of the mass and the sin of the angle α with the horizontal.
h = m * sin(α)
h = 3300 kg * sin(29°)
Step 2: Determine the final kinetic energy just before stopping.
The final kinetic energy is given by the formula: KE = (1/2)mv^2, where m is the mass and v is the final velocity.
Step 3: Calculate the work done by friction during skidding.
The work done by friction is equal to the force of friction multiplied by the distance.
Work = Force * Distance
The force of friction can be calculated using the formula: F = µN, where µ is the coefficient of kinetic friction and N is the normal force. The normal force is equal to the weight of the concrete mixer, which is given by the formula: N = mg.
Step 4: Equate the work done by friction to the change in kinetic energy.
Since the work done by friction is equal to the change in kinetic energy, we can write the equation: Work = ΔKE.
Step 5: Solve for the final velocity.
Rearrange the equation Work = ΔKE to solve for the final velocity v.
Once we have the final velocity, we can convert it from m/s to km/h by multiplying it by 3.6, which is the conversion factor.
I will now perform the calculations for you.