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U.S. Forest Service ("Limber Pine Dwarf Mistletoe", Forest Insect and Disease Leaflet 171, 1999), dwarf mistletoe is a parasitic plant that grows on the branches of pine trees.  The mistletoe extracts its water and nutrients directly from the tree.  One rare aspect of dwarf mistletoe is its seed dispersal mechanism.  Rather than relying on birds or wind to spread seeds from pine tree to pine tree, mature mistletoe fruit literally explodes (as a result of extreme water pressure within the fruit).  The explosion hurles the seed away from the pine tree.  The seeds are coated with a sticky substance which causes them to adhere to whatever they hit.  Ideally, the seed hits another nearby pine tree and begins to sprout.

The seed dispersal mechanism has been studied by T.E. Hinds and F.G. Hawksworth ( _Science_, Vol. 148, No. 3669 (Apr. 23 1965), pp. 517-519) by means of high-speed photography.  They find that _Arceuthobium cyanocarpum_ (the variety shown in the picture above) ejects is seeds with a speed of about 2100 cm/s.

  

Suppose that a certain dwarf mistletoe fruit expels a seed with a velocity of 2100 cm/s directed

 at 30° above the horizontal.  Suppose further that the seed hits another tree at exactly the same height that it was launched from.  Neglecting air resistance (*note:* neglecting air resistance is a poor assumption in this case) how far horizontally is the landing site displaced from the launch site?  

System:  The seed is treated as a point particle.

Models:  Projectile motion, assuming [One-Dimensional Motion with Constant Velocity|1-D Motion (Constant Velocity)] in the horizontal direction and [One-Dimensional Motion with Constant Acceleration|1-D Motion (Constant Acceleration)] in the vertical direction.

Approach:  We first sketch the situation and define a coordinate system.  

PICTURE

We choose the launch point to have the coordinates _x_ = 0 m, _y_ = 0 m and we choose to make _t_ = 0 s at the instant of launch.  We are now almost ready to summarize our givens, but we first have to deal with the velocity.  When putting the information given in the problem into a form suitable for use in our equations, we must break up all vector quantities into their _x_ and _y_ components.  For the initial velocity, this is done by constructing a vector triangle:

Using this triangle, we can see that:

{latex}\begin{large}\[ v_{x} = v \cos \theta = \mbox{18.2 m/s} \]\[ v_{y,{\rm i}} = v \sin \theta = \mbox{10.5 m/s}\]\end{large}{latex}

We can now state our givens:

{panel:title=givens}{latex}\begin{large}\[ t_{\rm i} = \mbox{0 s}\]\[x_{\rm i} = \mbox{0 m}\] \[y_{\rm i} = \mbox{0 m}\]\[y = \mbox{0 m} \]\[ v_{x} = \mbox{18.2 m/s} \] \[ v_{y,{\rm i}} = \mbox{10.5 m/s} \]\[a_{y} = -\mbox{9.8 m/s}^{2}\]\end{large}{latex}{panel}

Our end goal is to determine _x_, which will tell us the horizontal displacement of the seed during its flight.  As usual, however, we must first find the time by using the _y_ direction.  The most direct approach is to use the Law of Change:

{latex}\begin{large}\[ y = y_{\rm i} + v_{y,{\rm i}} (t-t_{\rm i}) + \frac{1}{2}a_{y}(t-t_{\rm i})^{2} \] \end{large}{latex}

which, after substituting zeros, can be solved to give:

{latex}\begin{large}\[ t = \mbox{0 s} \qquad \mbox{or}\qquad t = -\frac{2 v_{y}}{a_{y}} = \mbox{2.14 s} \] \end{large}{latex}

We can now solve the _x_ direction Law of Change:

{latex}\begin{large} \[ x = x_{\rm i} + v_{x} (t-t_{\rm i}) = -\frac{2 v_{x} v_{y}}{a_{y}} = \mbox{39 m} \] \end{large}{latex}

{tip}A good check is to think about whether this answer is _reasonable_.  Given what you know about pine trees, does a 39 m range seem large enough to make it likely the seed will hit another tree?{tip}