A roller coaster reaches the top of the steepest hill with a speed of 7.0 km/h . It then descends the hill, which is at an average angle of 44 ∘ and is 30.0 m long.
Estimate its speed when it reaches the bottom. Assume μk=0.18.
To estimate the speed of the roller coaster when it reaches the bottom of the hill, we can use the principle of conservation of energy. We'll need to consider the potential energy at the top and the kinetic energy at the bottom, taking into account any loss of energy due to friction.
The potential energy at the top of the hill can be calculated using the formula:
PE = mgh
where m is the mass of the roller coaster, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height of the hill.
To calculate the height of the hill, we'll need to use trigonometry. The height can be found using the equation:
h = l*sin(θ)
where l is the length of the hill and θ is the angle of inclination.
Substituting the given values, we have:
h = 30.0 m * sin(44°) ≈ 21.86 m
Now, we can calculate the potential energy at the top of the hill:
PE = m * g * h = m * 9.8 m/s² * 21.86 m
Next, we need to calculate the kinetic energy at the bottom of the hill. The law of conservation of energy states that the total energy at the top equals the total energy at the bottom, neglecting friction. So:
PE = KE
Since the roller coaster starts at rest at the top of the hill, its initial kinetic energy is zero. Therefore, the potential energy at the top of the hill equals the kinetic energy at the bottom:
PE = KE
m * 9.8 m/s² * 21.86 m = (1/2) * m * v^2
where v is the velocity of the roller coaster at the bottom of the hill.
Simplifying the equation and solving for v, we get:
v = sqrt(2 * 9.8 m/s² * 21.86 m)
Now, we can substitute the given values and calculate the estimated speed:
v = sqrt(2 * 9.8 m/s² * 21.86 m) ≈ 19.07 m/s
Finally, we convert the speed from m/s to km/h:
19.07 m/s * (3.6 km/h)/(1 m/s) ≈ 68.65 km/h
Therefore, the estimated speed of the roller coaster when it reaches the bottom of the hill is approximately 68.65 km/h.