You stand at the end of a long board of length L. The board rests on a frictionless frozen surface of a pond. You want to jump to the opposite end of the board. What is the minimum take-off speed v measured with respect to the pond that would allow you to accomplish that? The board and you have the same mass m.
Use g for the acceleration due to gravity
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.
To find the minimum take-off speed v that would allow you to jump to the opposite end of the board, you can use the principle of conservation of momentum and the principle of conservation of energy.
Step 1: Conservation of Momentum
- Start by considering the momentum before and after the jump.
- Initially, you and the board are at rest, so the total initial momentum is zero.
- If you can jump to the opposite end of the board, you would have a velocity v relative to the pond, while the board moves in the opposite direction with a velocity -v relative to the pond.
- Since momentum is conserved, the total final momentum should also be zero.
- The momentum of the board would be -m(-v) = mv, and your momentum would be mv.
- Therefore, mv + mv = 0 ⇒ 2mv = 0.
Step 2: Conservation of Energy
- Now, use the conservation of energy to determine the minimum take-off speed v.
- Initially, you and the board are at rest, so the initial kinetic energy is zero.
- The final kinetic energy is the sum of your kinetic energy and that of the board.
- Your kinetic energy is given by K1 = (1/2)mv² and the board's kinetic energy is K2 = (1/2)m(-v)² = (1/2)mv².
- The total final kinetic energy is K1 + K2 = (1/2)mv² + (1/2)mv² = mv².
- Since energy is conserved, the total final kinetic energy should be equal to the initial kinetic energy, which is zero.
- Therefore, mv² = 0.
Step 3: Combining the Results
- From step 1, we found that 2mv = 0.
- From step 2, we found that mv² = 0.
- Combining these equations, we can see that if 2mv = 0 and mv² = 0, then v = 0.
Conclusion:
- The minimum take-off speed v that would allow you to jump to the opposite end of the board is zero.