A 64.1-kg athlete leaps straight up into the air from a trampoline with an initial speed of 7.4 m/s. The goal of this problem is to find the maximum height she attains and her speed at half maximum height.
Write the general equation for energy conservation and solve for the velocity at half the maximum height. Substitute and obtain a numerical answer.
Write a general equation for energy conservation in this case and solve for the maximum height. Substitute and obtain a numerical answer.
To solve this problem, we will use the principle of energy conservation. In this case, the athlete's initial kinetic energy will be completely converted to potential energy at the maximum height.
The general equation for energy conservation is:
Initial kinetic energy + Initial potential energy = Final kinetic energy + Final potential energy
Let's break down the equation step by step:
1. Initial kinetic energy (KEi): The athlete's initial kinetic energy is given by the equation KEi = (1/2)mv^2, where m is the mass (64.1 kg) and v is the initial velocity (7.4 m/s).
2. Initial potential energy (PEi): Since the athlete is on the ground at the start, her initial potential energy is zero (PEi = 0).
3. Final kinetic energy (KEf): At the maximum height, the athlete's final kinetic energy will be zero since she momentarily comes to a stop (KEf = 0).
4. Final potential energy (PEf): The final potential energy can be calculated using the equation PEf = mgh, where m is the mass (64.1 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the maximum height we want to find.
Now, we can set up the equation using the above information:
(1/2)mv^2 + 0 = 0 + mgh
Substituting the numerical values:
(1/2)(64.1 kg)(7.4 m/s)^2 = (64.1 kg)(9.8 m/s^2)h
Simplifying the equation:
(1/2)(64.1 kg)(54.76 m^2/s^2) = (627.98 kg·m^2/s^2)h
Hence,
1668.238 kg·m^2/s^2 = (627.98 kg·m^2/s^2)h
Now, we can solve for the maximum height (h):
h = 1668.238 kg·m^2/s^2 / (627.98 kg·m^2/s^2)
Calculating this:
h ≈ 2.66 m (rounded to two decimal places)
So, the maximum height that the athlete reaches is approximately 2.66 meters.
To find the speed at half the maximum height, we need to solve for the velocity at that point. At half the maximum height, the potential energy is half the final potential energy.
PEf/2 = (1/2) mgh
Substituting the values:
(1/2)(64.1 kg)(9.8 m/s^2)h/2 = (1/2)(64.1 kg)(7.4 m/s)^2
Simplifying the equation:
(1/2)(627.98 kg·m^2/s^2)h/2 = (1/2)(1604.44 kg·m^2/s^2)
Now we can solve for h:
h = [(1/2)(1604.44 kg·m^2/s^2)] / [(1/2)(627.98 kg·m^2/s^2)]
Calculating this:
h ≈ 2.03 m (rounded to two decimal places)
So, the speed at half the maximum height is approximately 2.03 meters per second.