Suppose the roller coaster in Fig. 6-41 (h1 = 34 m, h2 = 10 m, h3 = 30) passes point 1 with a speed of 1.60 m/s. If the average force of friction is equal to one fourth of its weight, with what speed will it reach point 2? The distance traveled is 60.0 m.
without the figure...
To solve this problem, we can use the principle of conservation of mechanical energy. In the absence of non-conservative forces, the total mechanical energy of the roller coaster is conserved throughout its motion.
The total mechanical energy at any point on the roller coaster is the sum of its kinetic energy (KE) and its gravitational potential energy (PE). Mathematically, this can be expressed as:
ME = KE + PE
We can calculate the initial mechanical energy at point 1 and the final mechanical energy at point 2, and equate them to find the final speed at point 2.
1. Initial mechanical energy at point 1 (ME₁):
At point 1, the roller coaster has a speed of 1.60 m/s. Therefore, the initial kinetic energy (KE₁) at point 1 is given by:
KE₁ = 0.5 * m * v₁²
where m is the mass of the roller coaster and v₁ = 1.60 m/s.
The initial potential energy (PE₁) at point 1 is given by:
PE₁ = m * g * h₁
where g is the acceleration due to gravity (approximately 9.8 m/s²) and h₁ = 34 m.
The initial mechanical energy (ME₁) at point 1 is the sum of KE₁ and PE₁.
2. Final mechanical energy at point 2 (ME₂):
We can calculate the final potential energy (PE₂) at point 2 using the height h₂ = 10 m:
PE₂ = m * g * h₂
The final mechanical energy (ME₂) at point 2 is the sum of KE₂ (which we need to find) and PE₂.
Now, we can equate the initial mechanical energy (ME₁) to the final mechanical energy (ME₂) and solve for the final speed (v₂) at point 2.
ME₁ = ME₂
(0.5 * m * v₁²) + (m * g * h₁) = (0.5 * m * v₂²) + (m * g * h₂)
Next, we need to substitute the given values:
v₁ = 1.60 m/s
h₁ = 34 m
h₂ = 10 m
g ≈ 9.8 m/s² (acceleration due to gravity)
After plugging these values into the equation, we can solve for v₂.