An athlete with mass (m) running at speed (v) grabs a light rope that hangs from a ceiling of height H and swings to a maximum height of h₁. In another room with a lower ceiling of height H/2, a second athlete with mass 2m running at the same speed (v) grabs a light rope hanging from the ceiling and swings to a maximum height of h₂. How does the maximum height reached by the two athletes compare and why?
v = sqrt (2gh)
h depends only on v (on earth anyway)
To compare the maximum heights reached by the two athletes, we need to consider the principle of conservation of mechanical energy. According to this principle, the total mechanical energy of a system remains constant, neglecting any external forces such as friction or air resistance.
Let's define the initial kinetic energy (KE) of each athlete as they grab the rope. Since they are both running at the same speed (v), their initial kinetic energy will be the same:
KE_initial = (1/2)mv²
Now let's analyze what happens after they grab the rope and start swinging. At the maximum height, the kinetic energy will be zero (as the athletes momentarily stop moving horizontally), and all the initial kinetic energy will be converted to gravitational potential energy (PE_gravity).
PE_gravity = mgh
Since we want to compare the maximum heights, we can set the potential energies equal for both athletes and solve for h₁ and h₂:
mgh₁ = (1/2)mv²
gh₁ = v²/2
h₁ = v²/(2g)
2mgh₂ = (1/2)(2mv)²
4mgh₂ = 2mv²
2gh₂ = v²
gh₂ = v²/2
h₂ = v²/(2g)
Comparing h₁ and h₂:
h₁/h₂ = (v²/(2g)) / (v²/(2g)) = 1
Therefore, the maximum heights reached by the two athletes are equal (h₁ = h₂) because the mass of the second athlete (2m) compensates for the lower ceiling height (H/2).
To compare the maximum heights reached by the two athletes, we can analyze the conservation of mechanical energy. In this scenario, the mechanical energy comprises the kinetic energy and the potential energy of the system.
Let's start by calculating the maximum height reached by the first athlete with mass (m). When the athlete grabs the light rope, all of their initial kinetic energy is converted into potential energy when reaching the maximum height.
The initial kinetic energy of the first athlete is given by K = 0.5 * m * v².
At the maximum height h₁, the potential energy is given by the expression P = m * g * h₁, where g is the acceleration due to gravity.
Since energy is conserved, the initial kinetic energy (K) is equal to the potential energy (P) when at the maximum height:
0.5 * m * v² = m * g * h₁.
Next, let's analyze the maximum height reached by the second athlete with mass (2m) in the room with a lower ceiling of height (H/2). Again, the initial kinetic energy is converted entirely into potential energy at the maximum height.
The initial kinetic energy of the second athlete is given by K = 0.5 * (2m) * v².
At the maximum height h₂, the potential energy is given by the expression P = (2m) * g * h₂.
Since energy is conserved, the initial kinetic energy (K) is equal to the potential energy (P) when at the maximum height:
0.5 * (2m) * v² = (2m) * g * h₂.
To compare the maximum heights, we can divide the equation for the first athlete's maximum height (h₁) by the equation for the second athlete's maximum height (h₂):
(0.5 * m * v²) / (0.5 * (2m) * v²) = (m * g * h₁) / ((2m) * g * h₂).
Simplifying the equation gives:
1 / 2 = h₁ / (2 * h₂).
Thus, the ratio of the maximum heights reached by the two athletes is 1:2. The first athlete reaches a maximum height that is half of the maximum height reached by the second athlete.
This difference arises because the potential energy is directly proportional to the height. Since the second athlete has twice the mass as the first athlete, they possess more potential energy and, therefore, reach a higher maximum height.
because KE is 1/2 m v^2, and PE is mgh2
then h2 depends on v^2 because the masses are the same before and after.
so h2 = 1/2 v^2/g in both cases, so h2 is independent of mass, or the original height of the ceiling.