Final answer:
To find Renaud Lavillenie's minimum running speed approaching the vault, we can use the principle of conservation of mechanical energy. By setting up an equation involving the initial potential energy, final kinetic energy, and final potential energy, we can determine the required velocity. Substituting the given values, we can calculate Lavillenie's minimum running speed.
Step-by-step explanation:
In order to find Renaud Lavillenie's minimum running speed approaching the vault, we can use the principle of conservation of mechanical energy. The initial potential energy of Lavillenie at the starting line is equal to the sum of his kinetic energy and potential energy when he passes over the bar. We can set up the equation:
Initial Potential Energy = Final Kinetic Energy + Final Potential Energy
At the starting line, Lavillenie's potential energy is 0, and his kinetic energy is given by 0.5 * mass * velocity^2. When he passes over the bar, his potential energy is m * g * height, where m is the mass of Lavillenie and g is the acceleration due to gravity. His final kinetic energy is given by 0.5 * mass * (velocity + 1.35)^2 (since he was moving at 1.35 m/s as he passed over the bar).
Setting up the equation and solving for velocity gives:
0 = 0.5 * mass * velocity^2 + mass * g * height - 0.5 * mass * (velocity + 1.35)^2
0 = 0.5 * mass * velocity^2 + mass * g * height - 0.5 * mass * (velocity^2 + 2.7 * velocity + 1.8225)
0 = 0.5 * mass * velocity^2 - 0.5 * mass * velocity^2 - mass * g * height + 0.5 * mass * 2.7 * velocity + 0.5 * mass * 1.8225
Mass cancels out:
0 = 1.35 * velocity + 1.8225 - g * height
Solving for velocity:
1.35 * velocity = g * height - 1.8225
velocity = (g * height - 1.8225) / 1.35
Substituting the values for g = 9.8 m/s^2 and height = 6.16 m:
velocity = (9.8 * 6.16 - 1.8225) / 1.35 = 43.367m/s
The Question is incomplete. The complete question is
Renaud Lavillenie of France holds the current men's pole-vaulting record for his 2014 vault over a bar set 6.16 m above the floor. Find Renaud's minimum running speed (in m/s) approaching the vault, assuming he was moving at 1.35 m/s as he passed over the bar at the top of his vault. Neglect air resistance and any energy absorbed by the pole.