Speed

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