Problem 74 - Brief Applied Calculus

Question

ATHLETICS: Juggling If you toss a ball \(h\) feet straight up, it will return to your hand after \(T(h)=0.5 \sqrt{h}\) seconds. This leads to the juggler's dilemma: Juggling more balls means tossing them higher. However, the square root in the above formula means that tossing them twice as high does not gain twice as much time, but only \(\sqrt{2} \approx 1.4\) times as much time. Because of this, there is a limit to the number of balls that a person can juggle, which seems to be about ten. Use this formula to find: a. How long will a ball spend in the air if it is tossed to a height of 4 feet? 8 feet? b. How high must it be tossed to spend 2 seconds in the air? 3 seconds in the air?

Step-by-Step Solution

Answer

a. 1 second for 4 feet; 1.414 seconds for 8 feet. b. 16 feet for 2 seconds; 36 feet for 3 seconds.

Step 1: Calculate Time for 4 Feet Height

Using the formula for time, \( T(h) = 0.5 \sqrt{h} \), substitute \( h = 4 \) feet to find the time: \[ T(4) = 0.5 \times \sqrt{4} = 0.5 \times 2 = 1 \text{ second} \].

Step 2: Calculate Time for 8 Feet Height

Substitute \( h = 8 \) feet into the formula \( T(h) = 0.5 \sqrt{h} \): \[ T(8) = 0.5 \times \sqrt{8} = 0.5 \times 2.828 = 1.414 \text{ seconds} \].

Step 3: Calculate Height for 2 Seconds Air Time

We need to find \( h \) such that \( T(h) = 2 \) seconds. Rearrange the formula \( 0.5 \sqrt{h} = 2 \), giving us \( \sqrt{h} = 4 \). Squaring both sides, we find \( h = 16 \) feet.

Step 4: Calculate Height for 3 Seconds Air Time

Set \( T(h) = 3 \) seconds. Rearrange the formula: \( 0.5 \sqrt{h} = 3 \), therefore, \( \sqrt{h} = 6 \). Squaring both sides gives \( h = 36 \) feet.

Key Concepts

Vertical Motion Equations

Vertical motion equations help us understand how objects move when tossed in a vertical direction under the influence of gravity. When a ball is thrown upward, the motion can be described mathematically by considering its initial velocity and the effect of gravity. These equations are essential for calculating things like velocity, time of flight, and height. - Motion in vertical direction is influenced by gravity which constantly pulls objects back toward the earth.- For objects thrown up, they slow down until they reach maximum height and then accelerate back down. The basic components of these equations include:- **Initial velocity**: The speed at which the object is thrown. - **Time**: Duration the object is in motion. - **Height or distance**: How far the object travels vertically. - **Acceleration due to gravity**: On Earth, this is approximately \(9.8 \, \text{m/s}^2\), but we usually consider it as \(h = ext{initial velocity} \, \times \, \text{time} - \frac{1}{2} \, g \, \times \, ext{time}^2\). In the juggling problem, we use a simplified form to determine time based on height by using the vertical motion equation in a rearranged manner.

Square Root Function

A square root function is an important mathematical concept used in various physics equations, especially in motion problems. It is symbolized by \( \sqrt{} \) and is the inverse of squaring a number. The square root tells you what number, multiplied by itself, equals the original number.- In math terms: If \( x = \sqrt{y} \), then \( x^2 = y \). - This function produces only the non-negative root due to its definition.In our juggling scenario, the square root function is used to calculate the time a ball spends in the air. By substituting the height into the equation \( T(h) = 0.5 \sqrt{h} \), you can find how long it takes for the ball to reach the highest point and return to your hand.- For example, by substituting \( h = 4 \) into the function, you calculate: \( T(4) = 0.5 \times \sqrt{4} = 1 \) second.The use of the square root function here simplifies the physical problem by directly linking height and time, taking into account the effect of gravity.

Time of Flight Calculations

Time of flight represents the total duration an object spends in the air from the moment it is thrown until it returns to the starting point. Calculating this is crucial in many sports and physics problems, such as in the athletics juggling example.To determine time of flight, you use the relationship between height and time, as shown in the problem with the formula \( T(h) = 0.5 \sqrt{h} \). This formula simplifies the calculation by using a fixed multiplier coupled with the square root of the height.- **Example Calculations**: - For a height of 8 feet, the calculation is: \( T(8) = 0.5 \times \sqrt{8} \approx 1.414 \) seconds. - If you want a ball to be in the air for exactly 2 seconds, rearrange the formula to solve for \( h \): \( 2 = 0.5 \sqrt{h} \). This gives you \( \sqrt{h} = 4 \) and squaring gives \( h = 16 \) feet.Understanding this calculation allows jugglers to predict and plan the timing of their throws more effectively, maintaining rhythm and increasing the number of objects they can juggle.