Final answer:
To break the board, the hand must have kinetic energy equivalent to the work required to bend and break the board. Given the applied force and deflection, this corresponds to a hand speed of approximately 6.7 m/s.
Step-by-step explanation:
To break a pine board with a hand, we need to consider the transfer of kinetic energy from the hand to the potential energy required to bend and break the board. We are given that it takes an applied force of 870 N to break the board at a deflection of 1.3 cm, and we want to find out how fast the hand must be moving if it has a mass of 0.50 kg. The energy required to break the board is the work done by this force over the given deflection distance, which can be calculated using the equation W = F × d.
Calculating the work done, we get W = 870 N × 0.013 m = 11.31 J. This is the kinetic energy the hand must have just before impact. Using the kinetic energy formula KE = ½ mv^2, where m is the mass of the hand and v is its velocity, we can solve for v to find the required speed.
Setting kinetic energy equal to the work done to break the board and solving for v gives ½ × 0.50 kg × v^2 = 11.31 J. Solving for v, we find v ≈ 6.7 m/s. Thus, to break the board with a blow from the hand, the hand must be moving at a speed of approximately 6.7 m/s.