Question

Two people are floating in a hot air balloon 200 feet above a lake. One person tosses out a coin with an initial velocity of 20 feet per second. One second later, the balloon is 20 feet higher and the other person drops another coin (see figure). The position equation for the first coin is \(s=-16 t^{2}+20 t+200\) and the position equation for the second coin is \(s=-16 t^{2}+220 .\) Which coin will hit the water first? (Hint: Remember that the first coin was tossed one second before the second coin was dropped.)

Short answer

After performing the calculations in the steps outlined, we find that the first coin is the one that hits the water first.

Step-by-step solution

Solve the Quadratic Equation for the First Coin

Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \(a = -16\), \(b = 20\), and \(c = 200\). This gives two solutions - disregard the negative solution as time can't be negative. Calculate the positive root.

Solve The Quadratic Equation for the Second Coin

Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \(a = -16\), \(b = 0\), and \(c = 220\). This also gives two solutions - disregard the negative solution as time can't be negative. Calculate the positive root and remember to add 1 because this coin was dropped a second later.

Compare the Times of Both Coins Hitting the Water

Compare the times calculated in Steps 1 and 2. The coin that hits the water first is the one with the lesser time.

Key concepts

Projectile Motion

Projectile motion describes the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. This type of motion involves both horizontal and vertical displacement.
In the exercise, the coin thrown from the hot air balloon exemplifies projectile motion. The initial velocity and the force of gravity determine its path, which typically follows a parabolic trajectory. Understanding the parabola's properties and how its vertex relates to its maximum height is key. This trajectory results from resolving the object's initial velocity into horizontal and vertical components. The horizontal motion proceeds at constant speed, while the vertical motion involves uniform acceleration due to gravity.

• Horizontal velocity remains constant.
• Vertical motion is uniformly accelerated by gravity.
• The trajectory forms a parabola.

Recognizing these components helps not only to solve problems like the one presented but also to understand the wider applications of projectile motion in physics.

Physics Applications

Physics applications of projectile motion are vast, extending to various fields and phenomena. In sports, for example, understanding the physics of projectile motion can help calculate the best angle and speed for a basketball throw, a football kick, or a tennis serve.

In the problem at hand, calculating when the coins hit the water involves applying principles of trajectory and free fall. This includes determining when their vertical position reaches zero, thus hitting the water. The practical use of the quadratic equation in this scenario showcases how physics principles apply to our everyday observations and actions.

• Seen in sports for optimizing throws and kicks.
• Used to predict how objects move under gravity.
• Forms the basis for designing projectiles and launch systems.

By understanding the physics behind projectile motion, we bridge the gap between abstract mathematical models and tangible real-world applications.

Mathematical Problem Solving

Mathematical problem solving is crucial when approaching physics-related challenges, such as determining when objects in projectile motion will land. Quadratic equations, like those used in the exercise, are essential tools in this process.

The problem-solving pathway often involves identifying known variables, setting up equations based on those variables, and logical reasoning to find solutions. For the coin exercise, applying the quadratic formula helps determine the time each coin takes to reach the lake.

• Identify and define variables.
• Set up the relevant equations or formulas.
• Solve for unknowns using appropriate mathematical methods.

Beyond numerical answers, such problem-solving teaches you how to dissect a problem, employ logical strategies, and derive conclusions, skills that are valuable beyond physics.