Problem 73 - College Algebra and Calculus: An Applied Approach

Question

Assume that air resistance is negligible, which implies that the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) is a reasonable model Olympic Diver The high-dive platform in the Olympics is 10 meters above the water. A diver wants to perform an armstand dive, which means she will drop to the water from a handstand position. How long will the diver be in the air? (Hint: 1 meter \(\approx 3.2808\) feet)

Step-by-Step Solution

Answer

The diver will be in the air for approximately 1.43 seconds.

Step 1: Convert Initial Height from Meters to Feet

The initial height \(s_0\) of the falling object is given in meters. However, the equation uses feet, so we need to convert the height to feet before we can use the equation. Since 1 meter \(\approx 3.2808\) feet, we multiply 10 meters by equation, i.e., \(s_0 = 10 * 3.2808 = 32.808\) feet.

Step 2: Substitute into the Equation

We can now substitute these values into the equation. The initial velocity \(v_0\) is \(0\) (because the diver is 'dropped'), and the final position \(s\) is \(0\) (when the diver hits the water). This provides us with the equation \(0 = -16t^{2} + 0 + 32.808\).

Step 3: Solve for Time

By rearranging and simplifying the equation, we have \(16t^{2} = 32.808\). Divide by \(16\) on both sides to solve for \(t^2\), which gives \(t^{2} = 32.808 / 16 = 2.05\). Lastly, take the square root on both sides to solve for \(t\), which is \(\sqrt{2.05} = 1.43\) seconds. Hence, the diver will be in the air for approximately 1.43 seconds.

Key Concepts

Quadratic Equation

The quadratic equation is a vital tool in analyzing projectile motion. It helps us determine key values such as time, distance, and height within motion scenarios. For projectile motion, the equation is typically structured as \[ s = -16t^{2} + v_{0}t + s_{0} \]where:

• \(s\) represents the final position or height.
• \(t\) is the time in seconds.
• \(v_{0}\) is the initial velocity, which can be zero if the object starts from rest.
• \(s_{0}\) is the initial height of the object.

To solve the equation for a specific variable, like time, we substitute known values. For example, knowing the initial height \(s_{0}\) and that the final position \(s\) is the water level (or zero), you can solve for the time \(t\). First calculate \(-16t^{2} = s - s_{0}\), then simplify to find \(t^{2}\) and finally take the square root to find \(t\). These steps help predict how long a diver will be in the air before reaching the water.

Conversion of Units

Converting units is often necessary in physics problems to ensure consistency. In this problem, the initial height of the platform was given in meters, but the physical model we used assumed the height in feet. To convert from meters to feet, use the conversion factor:
1 meter \(\approx 3.2808\) feet.
Multiply the initial height of 10 meters by 3.2808 to obtain 32.808 feet. This calculated height is then used as \(s_{0}\) in the quadratic equation. Consistent unit usage allows for accurate calculations when solving physics problems. Such conversions are common in physics, ensuring that all quantities are in compatible units and reducing errors.

Physics in Sports

Physics plays a crucial role in understanding and analyzing various sports activities. When evaluating an Olympic diver’s performance, principles like projectile motion assist in predicting flight time and determining safe heights. For divers, the initial position greatly influences their descent and flight duration. Understanding these physics concepts aids coaches in providing better training.
Many sports involve projectiles, whether it’s a diver, a basketball, or a soccer ball. Codes of physics, such as gravity and initial velocity, govern their motion potential. Mastery of these principles can enhance strategic training, allowing athletes to optimize their performance by understanding how elements like angle, speed, and conditions affect successes and outcomes in sports.