The occupants of a car traveling at a speed of 30 m/s note that on a particular part of a road their apparent weight is 15% higher than their weight when driving on a flat road.
a. Is that part of the road a hill or a dip?
b. What is the vertical curvature of the road? Please show all the work needed to answer this question
I think it is a dip
and
v^2/R = 0.15*9.81
hill.
v^2/r=.85*9.8
solve for radius r, the radius of curvature at the top.
a. Well, if their apparent weight is 15% higher, it seems like they are experiencing more force pushing them down. So, I'd say it's safe to assume that they are driving on a hill. It's the kind of uphill battle they didn't sign up for!
b. Let's crunch some numbers for the vertical curvature! The apparent weight is related to the net force acting on the occupants, which is the difference between their actual weight and the normal force.
We'll start by finding the normal force when driving on a flat road. The normal force should equal the weight of the occupants, since there is no net force in the vertical direction.
Apparent weight = Normal force + Weight
When driving on a flat road, the apparent weight equals the normal force because there is no net force acting on the occupants.
15% higher weight = 100% weight + Normal force
To convert the percentage to a decimal, divide it by 100: 15%/100 = 0.15
1 + 0.15 = 1 + Normal force/Weight
1.15 = 1 + Normal force/Weight
Now, let's introduce the vertical curvature. When driving on this particular part of the road, there is a net force acting on the occupants in the vertical direction, due to the curvature.
We can represent the net force as M*v^2/R, where M is the mass of the car and occupants, v is the speed of the car, and R is the radius of curvature.
Substituting this into the equation:
1.15 = 1 + (M*v^2)/R
Simplifying further:
0.15 = (M*v^2)/R
Now, we have an equation relating the vertical curvature (R) of the road to the mass of the occupants (M) and the speed of the car (v). To solve for R, we rearrange the equation:
R = (M*v^2)/0.15
Just plug in the values for the mass (M) and speed (v) to find the vertical curvature (R) of the road.
And remember, even though math can be a bit bumpy, keep smiling like a pothole can't get you down!
To determine whether the part of the road is a hill or a dip, we need to analyze the change in apparent weight. Let's go step by step to find the answers and show all the work.
a. To identify whether the part of the road is a hill or a dip, we need to consider the relationship between the apparent weight and the actual weight.
Apparent weight can be calculated using the formula:
Apparent Weight = Actual Weight + Buoyant Force
When driving on a flat road, the apparent weight will be equal to the actual weight because there is no additional buoyant force acting on the car.
Given that the occupants of the car note a 15% increase in apparent weight, it means that the apparent weight is 15% higher than the actual weight.
Let's say the actual weight of the car occupants is W.
Therefore, the increase in apparent weight is 15% of W, which can be represented as 0.15W.
Apparent Weight = Actual Weight + 0.15W
Now, if the apparent weight is higher, it means that the buoyant force must be positive.
Thus, for a hill, the buoyant force points upward, while for a dip, the buoyant force points downward.
In this case, since the apparent weight is higher than the actual weight, the buoyant force must be pointing upward. Therefore, the road segment is a hill.
b. To find the vertical curvature of the road, we need to calculate the acceleration due to gravity (g) acting on the car while driving on the hill.
The apparent weight of an object on an inclined surface can be calculated using the formula:
Apparent Weight = actual weight * cos(θ)
Where θ is the angle of the incline.
Since the actual weight of the car occupants is W, and the apparent weight (taking slope into account) is 1.15W (15% higher than actual weight), we can set up the equation as follows:
1.15W = W * cos(θ)
Now, we can solve for θ by rearranging the equation:
cos(θ) = 1.15W / W
cos(θ) = 1.15
To find θ, we can take the inverse cosine (arccos) of both sides:
θ = arccos(1.15)
Using a calculator, we find:
θ ≈ 20.66 degrees
Therefore, the vertical curvature of the road is approximately 20.66 degrees.