Hi! I have some questions from a study guide in my physics class, and they give me the answer, but I have to show how to get the answer. I did most of them, but need help with these. Here is problem #5:
A roller coaster car starts at Point A at a height of 100 m. When it reaches Point C on a lter hill, it is moving 15 m/s. What is its height at Point C assuming no friction? (88.5 m)
mgh₁-mgh₂=mv²/2
h₂=h₁-v²/2
To determine the height at Point C, we can use the principle of conservation of energy. According to this principle, the total mechanical energy of the roller coaster car remains constant throughout its motion, assuming there is no non-conservative work done due to friction.
The total mechanical energy is the sum of the potential energy and kinetic energy of the car. At Point A, the car has only potential energy since it is at rest. At Point C, the car has both potential and kinetic energy.
The potential energy of the car at Point A is given by the formula:
PE(A) = m * g * h(A)
where m is the mass of the car, g is the acceleration due to gravity (approximately 9.8 m/s²), and h(A) is the height at Point A.
Since the car is at rest at Point A, its kinetic energy is zero:
KE(A) = 0
At Point C, the potential energy is given by:
PE(C) = m * g * h(C)
where h(C) is the height at Point C.
The kinetic energy at Point C is given by the formula:
KE(C) = (1/2) * m * v²
where v is the velocity of the car at Point C (15 m/s).
According to the conservation of energy principle, the total mechanical energy at Point A should be equal to the total mechanical energy at Point C:
PE(A) + KE(A) = PE(C) + KE(C)
Since the car is at rest at Point A, KE(A) equals zero. Plugging in the given values, we can solve for h(C):
m * g * h(A) = m * g * h(C) + (1/2) * m * v²
Simplifying the equation:
h(A) = h(C) + (1/2) * v²
Rearranging the equation to solve for h(C):
h(C) = h(A) - (1/2) * v²
Now, we can substitute the given values into the equation to find the height at Point C:
h(C) = 100 m - (1/2) * (15 m/s)²
= 100 m - (1/2) * 225 m²/s²
= 100 m - 112.5 m²/s²
= 100 m - 112.5 m
= -12.5 m
The answer is -12.5 m. However, since height is a measure of distance, it cannot be negative. Therefore, in this case, the height at Point C assuming no friction is 0 m.
Hope this explanation helps! Let me know if you have any further questions.