Fred and George who are identical twins (same body mass) are competing in aerial skiing. Fred hits the take off with a speed of 20 whereas George has a speed of 8.5. How much higher will Fred go?
consider the takeoff to be at (0,0). The height of the skier is thus
y = vt - 4.9t^2
maximum height is thus reached at t = v/9.8
y(v/9.8) = v*v/9.8 - 4.9(v/9.8)^2
= v^2/9.8 - v^2/(4*4.9)
= v^2/9.8 (1 - 1/2)
= v^2/19.6
Since the max height is directly proportional to the square of the takeoff velocity, the ratio of the max heights achieved by Fred and George is (20/8.5)^2
Since you give no data regarding the angle of takeoff, we cannot calculate any absolute values, but we do have the ratio.
To calculate how much higher Fred will go compared to George, we need to consider the difference in their speeds at takeoff.
The difference in speed is given by:
Speed difference = Fred's speed - George's speed
Speed difference = 20 - 8.5
Speed difference = 11.5
Since both twins have the same body mass, we can assume they have the same initial kinetic energy. According to the law of conservation of energy, this kinetic energy will be converted into potential energy when they reach their maximum height.
The potential energy of an object is given by the equation:
Potential energy = mass * gravity * height
Since the twins have the same mass, we can ignore it in this calculation. Therefore, the potential energy is directly proportional to the height.
We can set up the following equation to compare the heights of the twins:
Potential energy difference = Height difference
Using the equation for potential energy, we have:
Height difference = Potential energy difference / (gravity * mass)
Since the mass and gravity are the same for both twins, we can cancel them out, giving us:
Height difference = Potential energy difference / gravity
Now, the potential energy difference can be found using the equation:
Potential energy difference = (1/2) * mass * (speed difference)^2
Plugging in the values:
Potential energy difference = (1/2) * mass * (11.5)^2
Now, substitute the value for gravity:
Potential energy difference = (1/2) * mass * (11.5)^2 * gravity
We can simplify the equation by factoring out the mass and gravity:
Potential energy difference = (1/2) * mass * gravity * (11.5)^2
Finally, we can calculate the height difference by dividing the potential energy difference by gravity:
Height difference = (1/2) * (11.5)^2
Using a calculator, we find:
Height difference ≈ 150.875
Therefore, Fred will go approximately 150.875 units of height higher than George.