So I got this little sheet called
Shoot the Chute
To be sure you understand the preceeding question and answer, consider this one: A roller coaster is pulled to the top of a "shoot the chute" and allowed to roll down. For a bigger thrill you might wish the car to be going twice as fast at the bottom of the run. To make your wish come true, the chute should be
a) twice as high
b) three times as high
c) four times as high
d) five times as high
e) six times as high
Ok I know the answer is c but I have no idea why... can someone please explain this one to me...
Drop in potential energy = m g h
kinetic energy at bottom = (1/2) m v^2
m g h = (1/2) m v^2
v^2 = 2 g h
v = sqrt(2 g h)
if you multiply h by 4, you get twice v
To understand why option c) is the correct answer, we need to examine the concept of potential energy and kinetic energy.
When the roller coaster is at the top of the chute, it has potential energy due to its height above the ground. As it goes down the chute, the potential energy is converted into kinetic energy, which is the energy of motion. According to the law of conservation of energy, energy can neither be created nor destroyed, only transferred from one form to another.
Now, let's consider the scenario given in the question. If you want the car to be going twice as fast at the bottom of the run, it means you want it to have twice the kinetic energy compared to its initial state.
The kinetic energy (KE) of an object is directly proportional to its mass (m) and the square of its velocity (v). Mathematically, it can be represented as KE = 0.5 * m * v^2.
Since the mass of the roller coaster remains constant, to achieve double the kinetic energy, we need to double the square of its velocity.
The velocity of the roller coaster can be related to its height by the principles of mechanical energy conservation. The potential energy (PE) of an object at height h can be expressed as PE = m * g * h, where g is the acceleration due to gravity.
As the roller coaster goes down the chute, the potential energy is converted into kinetic energy, so we have the equation: PE = KE.
Using these equations, we can analyze the relationship between height and velocity. Let's assume the original height of the chute is h.
For option c) - Four times as high:
If the chute is four times as high, the potential energy at the top will also increase by a factor of four. Therefore, the kinetic energy at the bottom, which is converted from potential energy, will also increase by a factor of four. As a result, the velocity of the roller coaster at the bottom will be doubled, fulfilling the requirement mentioned in the question.
In contrast, for options a), b), d), and e), the increase in height would not lead to the desired doubling of the velocity at the bottom.
- Option a) - Twice as high: Doubling the height of the chute would only double the potential energy at the top, resulting in a √2 (square root of 2) increase in velocity at the bottom, not the desired doubling.
- Option b) - Three times as high: Tripling the height of the chute would triple the potential energy at the top, resulting in a √3 (square root of 3) increase in velocity at the bottom, not the desired doubling.
- Option d) and e): Similarly, quadrupling or quintupling the height of the chute would lead to a √4 (square root of 4) or √5 (square root of 5) increase in velocity, respectively, not the desired doubling.
Therefore, by considering the principles of potential energy, kinetic energy, and the conservation of energy, we can determine that option c) - Four times as high - is the correct answer in order to make the car go twice as fast at the bottom of the run.