# Application of quadratic function in kinematics

#### drop

##### New member
Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct

Part B

1) A golfer hits a nice iron shot and the ball's height above the ground is given by h(t) = -16t^2 + 80t, where t is the time in seconds since the ball was hit.

a) Evaluate h(4)

b) Explain the meaning of evaluating h(4) in the context of the problem.

My Answer: Evaluating h(4) will find the height in feet of the golf ball when it is 4 seconds after being struck by the golfer.

c) Determine the max height the golf ball reaches during the shot

d) Determine the time, t, when the maximum occurs.

#### MarkFL

Staff member

All correct.

As far as the maximum is concerned, there are several ways we can do this, and this doesn't even include using the calculus!

We know this parabola opens down, as the leading coefficient is negative. Thus, its global maximum will be at the vertex.

i) Factor:

$$\displaystyle h(t)=80t-16t^2=16t(5-t)$$

We see the roots are at $$\displaystyle t=0,\,5$$ and so the axis of symmetry, that value of $t$ on which the vertex lies, must be midway between the roots:

$$\displaystyle t=\frac{0+5}{2}=\frac{5}{2}$$

$$\displaystyle h\left(\frac{5}{2} \right)=16\cdot\frac{5}{2}\left(5-\frac{5}{2} \right)=(2\cdot5)^2=100$$

ii) Find the axis of symmetry without using the roots:

A quadratic of the form $$\displaystyle y=ax^2+bx+c$$ will have an axis of symmetry given by:

$$\displaystyle x=-\frac{b}{2a}$$

and so for the given function, we find the axis of symmetry is:

$$\displaystyle t=-\frac{80}{2(-16)}=\frac{5}{2}$$

Finding the value of the function at this value of $t$ is the same as above.

iii) Write the function in vertex form:

Completing the square, we find:

$$\displaystyle h(t)=-16t^2+80t=-16\left(t^2-5t+\left(\frac{5}{2} \right)^2 \right)+16\left(\frac{5}{2} \right)^2=-16\left(t-\frac{5}{2} \right)^2+100$$

And so we find the vertex is at $$\displaystyle \left(\frac{5}{2},100 \right)$$