Fred and George who are identical twins, same body mass, are competing in aerial skiing. Fred hits the take off with a speed of 15.8, whereas George has a speed of 8.3. How much higher will Fred go?
(1/2) m v^2 = m g h
h = (1/2) v^2 /g
h1 - h2 = (15.8^2 - 8.3^2) /(2 g)
To determine how much higher Fred will go compared to George, we can use the principle of conservation of energy. The total energy at the take-off of both skiers will be equal to the sum of their kinetic energy (due to their speed) and their potential energy (due to their height).
The formula for kinetic energy is:
KE = (1/2) * mass * velocity^2
Since Fred and George have the same body mass, their kinetic energy will be proportional to their velocities squared.
Let's calculate the kinetic energy for both skiers:
KE_Fred = (1/2) * mass * velocity_Fred^2
= (1/2) * mass * 15.8^2
KE_George = (1/2) * mass * velocity_George^2
= (1/2) * mass * 8.3^2
Since both Fred and George have the same body mass, we can ignore it when comparing their heights.
Now, let's find the difference in kinetic energy between Fred and George:
Delta_KE = KE_Fred - KE_George
= (1/2) * 15.8^2 - (1/2) * 8.3^2
Next, we need to find the equivalent difference in potential energy, as we want to know how much higher Fred will go. The potential energy is dependent on the height the skier reaches.
The formula for potential energy is:
PE = mass * acceleration due to gravity * height
Since we're comparing the heights, we can ignore the mass and the acceleration due to gravity.
Let's find the height difference by equating the potential energy difference to the calculated difference in kinetic energy:
Delta_PE = Delta_KE
Thus:
height_difference = Delta_PE / (mass * acceleration due to gravity)
= Delta_KE / (mass * acceleration due to gravity)
Now, we can substitute the value of Delta_KE and solve for the height difference:
height_difference = [(1/2) * 15.8^2 - (1/2) * 8.3^2] / (mass * acceleration due to gravity)
Please note that we need the values for the mass of the skiers and the acceleration due to gravity in order to calculate the actual height difference.