http://www.ck12.org Chapter 10. Polar and Parametric Equations
The vertical component also needs to include gravity and the starting height. The general equation for the vertical
component is:
y=^12 ·g·t^2 +t·v·sinθ+k
The constantgrepresents gravity,trepresents time,vrepresents initial velocity andkrepresents starting height. You
will explore this equation further in calculus and physics. Note that in this concept, most answers will be found and
confirmed using technology such as your graphing calculator.
Example A
A ball is thrown from the point (30, 5) at an angle of^49 πto the left at an initial velocity of 68f t/s.Model the position
of the ball over time using parametric equations. Use your graphing calculator to graph your equations for the first
four seconds while the ball is in the air.
Solution: The horizontal component isx=−t· 68 ·cos(^49 π) + 30 .Note the negative sign because the object is
traveling to the left and the +30 because the object starts at (30, 5).
The vertical component isy=^12 ·(− 32 )·t^2 +t· 68 ·sin(^49 π) + 5 .Note thatg=−32 because gravity has a force
of− 32 f t/s^2 and the+5 because the object starts at (30, 5).
Example B
When does the ball reach its maximum and when does the ball hit the ground? How far did the person throw the
ball?
Solution:To find when the function reaches its maximum, you can find the vertex of the parabola. Analytically
this is messy because of the decimal coefficients in the quadratic. Use your calculator to approximate the maximum
after you have graphed it. Depending on how small you make yourTste pshould find the maximum height to be
about 75 feet.
To find out when the ball hits the ground, you can set the vertical component equal to zero and solve the quadratic
equation. You can also use the table feature on your calculator to determine when the graph goes from having a
positive vertical value to a negative vertical value. The benefit for using the table is that it simultaneously tells you
thexvalue of the zero.