Problem 18 - A Graphical Approach to College Algebra

Question

Solve each problem. Hitting a Golf Ball A golf ball is hit so that its height \(h\) in feet after \(t\) seconds is $$ h(t)=-16 t^{2}+60 t $$ (a) What is the initial height of the golf ball? (b) How high is the golf ball after 1.5 seconds? (c) Find the maximum height of the golf ball.

Step-by-Step Solution

Answer

(a) 0 feet, (b) 54 feet, (c) 56.25 feet.

Step 1: Determine the Initial Height

To find the initial height of the golf ball, we need to evaluate the height function at the moment the ball is hit, which is when time \( t = 0 \). Substitute \( t = 0 \) into the function: \[h(0) = -16(0)^2 + 60(0) = 0.\]Therefore, the initial height of the golf ball is 0 feet.

Step 2: Calculate the Height After 1.5 Seconds

Given that the height function is \( h(t) = -16t^2 + 60t \), we need to find the height at \( t = 1.5 \) seconds. Substitute \( t = 1.5 \) into the function:\[h(1.5) = -16(1.5)^2 + 60(1.5).\]Calculate \( (1.5)^2 = 2.25 \) and then:\[h(1.5) = -16(2.25) + 60(1.5) = -36 + 90 = 54.\]So, the height of the golf ball after 1.5 seconds is 54 feet.

Step 3: Find the Maximum Height

The height as a function of time, \( h(t) = -16t^2 + 60t \), is a quadratic function, which forms a parabola opening downwards. The maximum point is at the vertex. For a quadratic \( at^2 + bt + c \), the vertex occurs at \( t = -\frac{b}{2a} \). Here, \( a = -16 \) and \( b = 60 \):\[t = -\frac{60}{2(-16)} = \frac{60}{32} = \frac{15}{8} = 1.875.\]We substitute \( t = 1.875 \) back into the height function:\[h(1.875) = -16(1.875)^2 + 60(1.875).\]Calculate \( (1.875)^2 = 3.515625 \) and then:\[h(1.875) = -16(3.515625) + 60(1.875) = -56.25 + 112.5 = 56.25.\]Thus, the maximum height of the golf ball is 56.25 feet.

Key Concepts

Projectile Motion

When a golf ball is hit into the air, it travels along a curve called a parabola. This is an example of **projectile motion**, which describes the path of any object thrown or propelled and affected by gravity. The height of the golf ball at any time can be described with a quadratic equation, like the one given: \[ h(t) = -16t^2 + 60t. \]

• The negative sign before the \( t^2 \) term indicates that the parabola opens downwards.
• The coefficient \(-16\) represents the acceleration due to gravity, which, in this equation, reduces the height over time.

Understanding projectile motion helps us calculate how high and how far objects will travel. In problems involving projectile motion, time \( t \), initial velocity, and gravity come together to determine the object's path.
It's important to remember that the only acceleration acting on the ball after it is hit is gravity, pulling it back towards the ground. This means the ball will rise to a certain point—its highest, most elevated point—before descending.

Vertex of a Parabola

The vertex of the parabola is a crucial point in understanding quadratic functions, particularly in contexts like projectile motion. In the equation for the golf ball’s height \( h(t) = -16t^2 + 60t \), the vertex represents the point where the golf ball reaches its maximum height.
To find this vertex, use the formula for the time to the vertex in a quadratic \( at^2 + bt + c \), which occurs at \( t = -\frac{b}{2a} \). Here, the values are:

• \( a = -16 \)
• \( b = 60 \)

Calculating gives: \[ t = -\frac{60}{2(-16)} = \frac{15}{8} = 1.875. \]This means the vertex is at 1.875 seconds, when the maximum height is achieved.
The vertex is always a turning point in the context of projectile motion. It marks where the initial upward motion due to velocity becomes totally counteracted by gravity.

Maximum Height

In the study of projectile motion, the **maximum height** is the highest point the object reaches above the starting point. For the golf ball example, this height is found at the vertex of the parabola.
Once the time when the ball reaches the vertex is calculated as \( t = 1.875 \) seconds, we use this time to find the height at that specific moment:\[ h(1.875) = -16(1.875)^2 + 60(1.875). \]After performing the calculations:

• \( (1.875)^2 = 3.515625 \)
• \( h(1.875) = -56.25 + 112.5 = 56.25 \)

Thus, the maximum height of the golf ball is 56.25 feet.
This maximum height is crucial as it tells us how far the ball can travel before gravity brings it back down. Understanding this concept can help improve strategies and predictions in various fields, from sports to physics.