Vectors Calaulus 1.16

PROBLEM 3   Jason Elam (the football kicker for the Denver Broncos) can kick a football with an initial velocity of 60 feet per second.  At what angle should the ball be kicked to maximize the horizontal distance that the ball travels before it lands on the ground?   (Use vectors please).

  Solution:

The acceleration is

        a(t)  =  -32 j 

Integrating gives

        v(t)  =  v_x i + (-32t + v_y) j  =  60cosq i + (-32t + 60sinq) j

Now integrate again to get

        r(t)  =  60t cosq i + (-16t² + 60sinq t) j

Notice that these constants are all zero since the ball starts at the origin.  The ball will reach its maximal horizontal distance when the j component equals 0.  We have

        -16t² + 60sinq t  =  0

        t  =  15/4 sinq

Now plug back into the i component and maximize

        h  =  60 (15/4 sinq) cosq   =  225/2 sin(2q)     

Now to maximize, we take the derivative and set it equal to zero

        h'  =  225 cos(2q)

This is zero when q  =  p/4.

  Jason Elam should punt the ball with an initial angle of p/4.

PROBLEM 4  Prove the following theorem:

Let r(t) be a differentiable vector valued function, then

        |(r x v) ^. a|  =  ||r'|| ||a_N|| |r ^. (T x N)|

Solution:

Since 

        v  =  ||r'|| T

we have

         |(r x v) ^. a|  =  |(r x ||r'||T) ^. a| =  |(r x ||r'||T) ^. (a_TT + a_NN)| 

        =  ||r'|| |(r x T) ^. a_TT + (r x T) ^. a_NN)|  

Since (r x T) is orthogonal to T, the first term is zero.  We get

        =  ||r'|| |(r x T) ^. a_NN)|  =   ||r'|| |a_N|  |(r x T) ^. N)|