Suppose the roller coaster in Fig. 6-41 (h1 = 30 m, h2 = 11 m, h3 = 20) passes point 1 with a speed of 2.10 m/s. If the average force of friction is equal to one fifth of its weight, with what speed will it reach point 2? The distance traveled is 50.0 m.
i think the answer is big breasts
To solve this problem, we can apply the principle of conservation of mechanical energy. This principle states that the total mechanical energy of a system remains constant as long as no external forces (except for conservative forces) act on the system.
In this case, the roller coaster is subjected to the force of gravity and the force of friction. However, the force of gravity is a conservative force, so it does not affect the total mechanical energy of the system. Only the force of friction needs to be considered.
The total mechanical energy of the system is given by the sum of the kinetic energy (KE) and the gravitational potential energy (PE) of the roller coaster.
At point 1, we have:
KE1 + PE1 = KE2 + PE2
The kinetic energy is given by the formula KE = (1/2) * m * v^2, where m is the mass of the roller coaster and v is its velocity.
The gravitational potential energy is given by the formula PE = m * g * h, where g is the acceleration due to gravity and h is the height.
Let's calculate the values for each point:
At point 1:
KE1 = (1/2) * m * v1^2 (where v1 = 2.10 m/s)
PE1 = m * g * h1 (where h1 = 30 m)
At point 2:
KE2 = (1/2) * m * v2^2 (we want to find v2)
PE2 = m * g * h2 (where h2 = 11 m)
Since the roller coaster does work against the force of friction, the work done by friction (∆W) can be calculated using the formula ∆W = f * d, where f is the force of friction and d is the distance traveled.
In this case, the average force of friction is equal to one fifth of its weight. Therefore, the force of friction can be calculated as f = (1/5) * m * g, where g is the acceleration due to gravity.
Let's calculate the value of the force of friction using this formula, and substitute it in the work-energy equation:
∆W = f * d = (1/5) * m * g * d
Now, since work is equal to the change in mechanical energy (∆W = PE2 - PE1), we can set up the equation:
PE2 - PE1 = (1/5) * m * g * d
Substituting the formulas for gravitational potential energy:
m * g * h2 - m * g * h1 = (1/5) * m * g * d
The mass and acceleration due to gravity (g) appear on both sides of the equation, so they cancel out:
g * (h2 - h1) = (1/5) * g * d
After simplifying:
h2 - h1 = (1/5) * d
Now we can solve for h2:
h2 = h1 + (1/5) * d
Substituting the given values:
h1 = 30 m
d = 50.0 m
h2 = 30 + (1/5) * 50.0 = 40.0 m
Finally, we can calculate v2 using the equation:
KE2 = (1/2) * m * v2^2
PE2 = m * g * h2
KE1 + PE1 = KE2 + PE2
(1/2) * m * v1^2 + m * g * h1 = (1/2) * m * v2^2 + m * g * h2
(1/2) * v1^2 + g * h1 = (1/2) * v2^2 + g * h2
Substituting the known values:
(1/2) * (2.10)^2 + 9.81 * 30 = (1/2) * v2^2 + 9.81 * 40.0
Simplifying the equation:
(1/2) * 2.10^2 + 9.81 * 30 = (1/2) * v2^2 + 9.81 * 40
Now, solve for v2 by rearranging the equation:
(1/2) * v2^2 = (1/2) * 2.10^2 + 9.81 * 30 - 9.81 * 40
v2^2 = 2.10^2 + 2 * 9.81 * (30 - 40)
v2^2 = 2.10^2 + 2 * 9.81 * (-10)
v2^2 = 2.10^2 - 2 * 9.81 * 10
v2^2 = 2.10^2 - 196.2
v2^2 = 4.41 - 196.2
v2^2 = -191.79
Since the result is negative, it means that the roller coaster does not reach point 2.
To solve this problem, we need to apply the conservation of energy principle. The total mechanical energy of the roller coaster is the sum of its potential energy and kinetic energy. We'll calculate the roller coaster's speed at point 2 by equating the initial mechanical energy at point 1 to the final mechanical energy at point 2.
First, let's calculate the roller coaster's mechanical energy at point 1. The initial mechanical energy (E1) is given by:
E1 = mgh1 + (1/2)mv1^2
Where:
m is the mass of the roller coaster
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h1 is the height at point 1
v1 is the speed at point 1
Next, let's calculate the roller coaster's mechanical energy at point 2. The final mechanical energy (E2) is given by:
E2 = mgh2 + (1/2)mv2^2
Where:
h2 is the height at point 2
v2 is the speed at point 2
Now, since the average force of friction is equal to one fifth of the roller coaster's weight, we can calculate the frictional force (Ff) as follows:
Ff = (1/5)mg
To find the force due to gravity (Fg), we can calculate it as:
Fg = mg
Since the net work done by all forces acting on the roller coaster is equal to the change in its mechanical energy, we can write:
(Ff - Fg)d = E2 - E1
Where:
d is the distance traveled
E2 - E1 is the change in mechanical energy
We can rearrange the equation to solve for v2:
v2 = sqrt((2/g)(E2 - E1) + v1^2)
Now let's calculate the values and solve the equation:
Given:
h1 = 30 m
h2 = 11 m
h3 = 20 m
v1 = 2.10 m/s
d = 50.0 m
Now let's calculate the masses using the heights h1, h2, and h3:
m = h1 / g
m = 30 m / 9.8 m/s^2
m ≈ 3.06 kg
Now let's calculate the roller coaster's mechanical energy at point 1 (E1):
E1 = mgh1 + (1/2)mv1^2
E1 = (3.06 kg)(9.8 m/s^2)(30 m) + (1/2)(3.06 kg)(2.10 m/s)^2
E1 ≈ 888.13 J
Now let's calculate the roller coaster's mechanical energy at point 2 (E2):
E2 = mgh2 + (1/2)mv2^2
E2 = (3.06 kg)(9.8 m/s^2)(11 m) + (1/2)(3.06 kg)(v2)^2
Now let's calculate the frictional force (Ff):
Ff = (1/5)mg
Ff = (1/5)(3.06 kg)(9.8 m/s^2)
Ff ≈ 5.99 N
Now let's calculate the force due to gravity (Fg):
Fg = mg
Fg = (3.06 kg)(9.8 m/s^2)
Fg ≈ 29.92 N
Now let's substitute the obtained values into the equation to find v2:
(sqrt((2/g)(E2 - E1) + v1^2) = v2
(sqrt((2/9.8 m/s^2)(E2 - 888.13 J) + (2.10 m/s)^2) = v2
Then, we can substitute the appropriate values and solve for v2.