Green 1 Alice Green Mr. Smith Physics 16 May 2017 Speed A problem that is challenging enough for an average college student is the following. What is the slowest initial speed a boy can throw a pebble over a structure which height and length are equal to H and L respectively? Assume that the initial position of the pebble is arbitrary in the horizontal direction and equals h in the vertical one (the vertical axis points upward and its origin is the ground level). The easy way of solving this problem consists in taking advantage of the broadlyknown fact that is the following: The object thrown at the angle φ with the horizontal above the surface that is not inclined lands at the farthest point if φ = 45°. Specifically, from symmetry considerations, it follows that the initial speed is the smallest when the pebble trajectory is the one depicted on Figure 1 and γ = 45°. At the point A (see Figure 1) the vertical component of the stone velocity is equal to vA sin γ whereas at the point B this component equals -vA sin γ. Green 2 Figure 1. The acceleration in the vertical direction is equal to -g where g is the free fall one. Hence, if tAB is the time of travel of the object from point A to point B, then one can write the following: TAB = 2vA sin γ . (1) g Since the horizontal component of the stone acceleration is zero, one can write the following: tAB vA cos γ = L. (2) From the equations (1) and (2), it follows that VA = (3) Lg since γ = 45°. Hence, from the energy conservation and the equation (3), it follows that the smallest initial speed v0 of the pebble that guarantees reaching the opposite side of the structure satisfies such equation: Green 3 From the equation (4), it follows that

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