A car starts rolling down a 1-in-8 hill (1-in-8 means that for each 8 m traveled along the road, the elevation change is 1 m).
A) How fast is it going when it reaches the bottom after traveling 90 m ? Assume an effective coefficient of friction equal to 0.10.
Well, we all know that going downhill can be quite a rollercoaster ride! So, let's calculate the speed of this adventurous car.
First, we need to find the height it has descended. Since for every 8 meters traveled the elevation change is 1 meter, we can divide 90 meters by 8 meters to find the number of elevation changes. So, 90 meters divided by 8 meters equals 11.25 elevation changes.
Now, we multiply the number of elevation changes by 1 meter to find the total height the car has descended. 11.25 elevation changes times 1 meter equals 11.25 meters.
Next, we need to consider the coefficient of friction. With an effective coefficient of friction of 0.10, it means that the car will experience some resistance from the surface.
Assuming no other forces are acting on the car such as air resistance, we can calculate its final velocity using the equation: v = sqrt(2gh), where v is the velocity, g is the acceleration due to gravity (approximated as 9.8 m/s^2), and h is the height.
Plugging in the values, v = sqrt(2 * 9.8 m/s^2 * 11.25 m) = sqrt(214.5 m^2/s^2) ≈ 14.6 m/s.
So, after traveling 90 meters down the hill, the car will be going at approximately 14.6 meters per second. That's quite a fast-track downhill ride!
To calculate the speed of the car when it reaches the bottom of the hill, we can use the principle of conservation of mechanical energy.
The potential energy at the top of the hill is converted into kinetic energy at the bottom. The potential energy can be calculated using the formula:
Potential energy = mass * acceleration due to gravity * height
Since the height is given by the ratio of 1-in-8, we can calculate the height as:
height = (1/8) * 90 = 11.25 m
The mass and acceleration due to gravity cancels out on both sides of the equation, thus we can write:
Potential energy = Kinetic energy
Now, let's calculate the potential energy:
Potential energy = m * g * height
where m is the mass of the car and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Next, we can calculate the kinetic energy using the formula:
Kinetic energy = (1/2) * m * v^2
where v is the velocity of the car when it reaches the bottom.
Setting the potential energy equal to the kinetic energy, we have:
m * g * height = (1/2) * m * v^2
We can cancel out the mass 'm' and rearrange the equation to solve for velocity 'v':
v^2 = 2 * g * height
Taking the square root of both sides:
v = sqrt(2 * g * height)
Now, substitute the values:
v = sqrt(2 * 9.8 * 11.25) ≈ 14.68 m/s
Therefore, the car is traveling at approximately 14.68 m/s when it reaches the bottom of the hill after traveling 90 m.
To calculate the speed of the car when it reaches the bottom of the hill, we can use the principles of energy conservation and the concept of work done.
Firstly, we need to find the change in potential energy (ΔPE) of the car as it moves downhill. The formula to calculate the change in potential energy is:
ΔPE = m * g * Δh
Where:
m = mass of the car
g = acceleration due to gravity (approximately 9.8 m/s²)
Δh = change in height
In this case, the change in height is given by 90 m (the distance traveled by the car). However, the hill has a slope of 1-in-8, which means that for every 8 m traveled along the road, the elevation changes by 1 m.
To find the actual change in height for 90 m, we need to convert it into the equivalent vertical distance. Since the slope is 1-in-8, we can divide 90 m by 8 to get the change in height:
Δh = (90 m) / 8 = 11.25 m
Now we can calculate the change in potential energy:
ΔPE = m * g * Δh
Next, we need to determine the work done by the friction force in opposing the car's motion. The formula to calculate work is:
Work = force * distance
The force of friction can be calculated using the formula:
force = coefficient of friction * normal force
The normal force is equal to the weight of the car, which can be calculated as:
weight = m * g
To find the force of friction, we can plug in the value of the coefficient of friction (0.10) and the weight of the car.
Finally, we can calculate the work done by friction over the distance traveled downhill (90 m):
Work = force * distance
To find the speed of the car at the bottom of the hill, we can equate the work done by friction to the change in potential energy:
Work = ΔPE
Since work is equal to force multiplied by distance, we have:
(force * distance) = ΔPE
Now we can solve for the force:
force = ΔPE / distance
To find the speed of the car when it reaches the bottom of the hill, we can use the principle of conservation of energy:
Initial energy (KE + PE) = Final energy (KE + PE)
At the top of the hill, the car has only potential energy (PE), and at the bottom of the hill, the car has both kinetic energy (KE) and potential energy (PE). The change in energy can be calculated as:
ΔPE = -ΔKE
Where ΔKE is the change in kinetic energy from top to bottom.
Since kinetic energy is given by the formula:
KE = (1/2) * m * v²
Where v is the velocity of the car at the bottom of the hill.
Now we can substitute the equations and solve for the velocity (v):
(1/2) * m * v² = ΔPE
Substitute the value of ΔPE:
(1/2) * m * v² = force * distance
Solve for velocity (v):
v² = (2 * force * distance) / m
v = √((2 * force * distance) / m)
Plug in the respective values for force, distance, and mass of the car to find the velocity.
on the slope, sinTheta=1/8
friction=.1*mg*cosTheta (figure out cosTheta from the sine)
force down the slope=mg*sinTheta
net force=mg*sinTheta-.1mg*cosTheta
net acceleation=force/mass= you do it.
vf^2=2*netAcceleration*distance