A water skier lets go of the tow rope upon leaving the end of a jump ramp at a speed of 17.9 m/s. As the drawing indicates, the skier has a speed of 11.1 m/s at the highest point of the jump. Ignoring air resistance, determine the skier's height H above the top of the ramp at the highest point.
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At the highest point of the jump, the velocity will have only a horizontal component: 11.1 m/s. Since that component is constant, the vertical component when leaving the ramp was
sqrt[(17.9)^2 - (11.1)^2] = 14.0 m/s
The initial vertical velocity component, Vyo, can be used to calculate the maximum height H. The time that the jumper spends going up is
T = Vy,0/g. The average velocity while going up is Vy,o/2. H is the product of those two numbers.
To determine the skier's height above the top of the ramp at the highest point, we can use the principles of conservation of energy. At the highest point of the jump, the skier's energy will be entirely in the form of gravitational potential energy.
At the highest point of the jump, the skier's speed is given as 11.1 m/s. Let's assume the skier's initial height at the top of the ramp is H.
The initial kinetic energy of the skier can be calculated using the formula:
KE_initial = (1/2) * m * v^2
where m is the mass of the skier and v is the initial speed.
The final gravitational potential energy of the skier at the highest point can be calculated using the formula:
PE_final = m * g * H
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
According to the law of conservation of energy, the initial kinetic energy of the skier is equal to the final gravitational potential energy. Therefore, we can set up the equation:
(1/2) * m * v_initial^2 = m * g * H
Simplifying the equation, we have:
(1/2) * v_initial^2 = g * H
Rearranging the equation to solve for H, we get:
H = (1/2) * v_initial^2 / g
Substituting the given values, we can calculate the skier's height H:
H = (1/2) * (11.1 m/s)^2 / 9.8 m/s^2
H = (1/2) * 123.21 m^2/s^2 / 9.8 m/s^2
H = (1/2) * 12.582
H ≈ 6.291 m
Therefore, the skier's height above the top of the ramp at the highest point is approximately 6.291 meters.
To determine the skier's height above the top of the ramp at the highest point, we can use the principle of conservation of energy.
At the highest point of the jump, the skier's speed is 11.1 m/s. We can assume that the skier's initial speed at the bottom of the ramp is 17.9 m/s.
The total mechanical energy of the skier at the bottom of the ramp is the sum of kinetic energy and gravitational potential energy:
E(bottom) = KE(bottom) + PE(bottom)
At the highest point of the jump, the total mechanical energy of the skier is the sum of kinetic energy and gravitational potential energy:
E(highest) = KE(highest) + PE(highest)
Since there is no air resistance, the total mechanical energy of the skier is conserved, meaning E(bottom) = E(highest).
The kinetic energy of an object is given by the formula KE = (1/2)mv^2, where m is the mass and v is the velocity (speed) of the object.
Assuming the mass of the skier remains constant, we can equate the kinetic energy at the bottom and highest points:
(1/2)mv(bottom)^2 = (1/2)mv(highest)^2
Simplifying the equation:
v(bottom)^2 = v(highest)^2
(17.9 m/s)^2 = (11.1 m/s)^2
320.41 = 123.21
Now that we have confirmed the conservation of kinetic energy, we can calculate the height H.
The gravitational potential energy of an object near the surface of the Earth is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height.
Equating the potential energies at the bottom and highest points:
mgh(bottom) = mgh(highest)
Since the mass cancels out on both sides:
gh(bottom) = gh(highest)
Simplifying the equation:
g × H(bottom) = g × H(highest)
H(bottom) = H(highest)
Therefore, the height above the top of the ramp, H, is the same at the bottom and highest points.
In conclusion, the skier's height above the top of the ramp at the highest point is the same as the height above the ramp at the bottom, which is given by the height of the ramp. Unfortunately, without the dimensions of the ramp or any other additional information, we cannot determine the exact height.