To determine the minimum take-off speed required to jump to the opposite end of the board, we can apply the principle of conservation of energy.
First, let's consider the initial state when you are standing at one end of the board. At this point, your potential energy is given by m * g * L, where m is the mass and g is the acceleration due to gravity. Since you're on a frictionless surface, there is no energy loss due to friction.
Now, when you jump to the opposite end of the board, you will reach maximum height and have zero velocity at that point. At this highest point, your kinetic energy is zero. All the initial potential energy is converted to potential energy at the highest point.
The potential energy at the highest point is given by m * g * (L/2), since you are halfway horizontally along the board. At this point, your kinetic energy is zero, so the total mechanical energy (potential energy + kinetic energy) is given by:
Potential Energy = m * g * (L/2)
Kinetic Energy = 0
Since the total mechanical energy is conserved, the initial potential energy must equal the sum of the potential energy and kinetic energy at the highest point. Therefore, we can equate these two quantities:
m * g * L = m * g * (L/2) + 0
Now, let's solve for the minimum take-off speed v:
mgL = m * g * (L/2)
gL = g * (L/2)
L = L/2
Dividing both sides by m and g:
L/2 = L/2
This shows us that the minimum take-off speed required to jump to the opposite end of the board is zero. This means you can achieve the jump without any initial take-off speed, as long as the board is frictionless and you have the same mass as the board.