Problem 43 - Precalculus

Question

Involve vertical motion and the effect of gravity on an object. Because of gravity, an object that is projected upward will cventually reach a maximum height and then fall to the ground. The equation that determines the height \(h\) of a projectile \(t\) seconds after it is shot upward is given by $$ h=\frac{1}{2} a t^{2}+v_{0} t+h_{0} $$ where \(a\) is the acceleration due to gravity, \(h_{0}\) is the initial height of the object at time \(t=0,\) and \(v_{0}\) is the initial velocity of the object at time \(t=0 .\) Note that a projectile follows the path of a parabola opening down, so \(a<0\) The Seatule Times reported a story on November \(18,2006,\) about a game of Scrabble played between a human and a computer. The best Scrabble player in the United States was pitted against a computer program designed to play the game. Remarkably, the human beat the computer in the best of two out of three games competition. The total points scored by both computer and the man for all three games was \(2591 .\) The difference between the first game's total and second game's total was 62 points. The difference between the first game's total and the third game's total was only 2 points. Determine the total number of points scored by both computer and the man for each of the three contests.

Step-by-Step Solution

Answer

First game: 885 points, second game: 823 points, third game: 883 points.

Step 1: Assign Variables for Game Scores

Let's denote the total points in the first game as \( x \). According to the problem, the total points of the second game would be 62 points less than the first, and the total points of the third game would be 2 points less than the first. Thus, we can express the points of the games as follows:- First game: \( x \)- Second game: \( x - 62 \)- Third game: \( x - 2 \)

Step 2: Write the Equation for Total Points

The total points for all three games is given as 2591. Therefore, sum the points of all three games and set the equation equal to 2591:\[x + (x - 62) + (x - 2) = 2591\]

Step 3: Simplify the Equation

Combine like terms in the equation:\[3x - 64 = 2591\]

Step 4: Solve for \( x \)

Add 64 to both sides of the equation to solve for \( x \):\[3x = 2655\]Divide both sides by 3:\[x = 885\]

Step 5: Calculate Scores for Each Game

Now that we know \( x = 885 \), substitute \( x \) back into the expressions for each game's total points:- First game: \( 885 \)- Second game: \( 885 - 62 = 823 \)- Third game: \( 885 - 2 = 883 \)

Step 6: Verify the Solution

Ensure that the sum of the three games equals the total reported score:\[885 + 823 + 883 = 2591\] The solution is correct as it satisfies the conditions given in the problem.

Key Concepts

Gravity

Gravity is a force that pulls objects toward the center of the Earth. When dealing with vertical motion, gravity is responsible for the downward acceleration of any object thrown upward. This accelerative force due to gravity is a constant on Earth, usually denoted by the symbol \( g \), with a value approximating \( 9.8 \text{ m/s}^2 \). This means that every second, the object's velocity changes by \( 9.8 \text{ m/s} \) downward.

In the context of projectile motion, gravity acts as the key determinant that influences how high an object will rise before beginning its descent. This ascent and eventual fall form the essence of vertical motion studies. For any object projected upwards, the initial upward force needs to counteract this constant pull of gravity until gravity gradually decreases the object's velocity to zero at its peak height. Afterward, gravity accelerates the object back downward.

Projectile Motion

Projectile motion refers to the movement of an object thrown into space, primarily influenced by gravity and its initial velocity. This object, which we call a projectile, follows a curved path known as a trajectory. In vertical motion, a projectile describes a simple up-and-down flight path.

Important aspects of projectile motion include:

• Initial Launch: High initial velocity results in a higher trajectory.
• Maximum Height: This is the highest point in the projectile's path.
• Time of Flight: Total duration of the projectile's journey.
• Impact Point: Where the projectile lands after completing its trajectory.

Understanding projectile motion involves analyzing how these components interact under the consistent influence of gravity. For a vertically launched projectile, the path is symmetrical, meaning the time going up is equal to the time coming down.

Quadratic Equations

In physics, especially while solving projectile motion problems, quadratic equations are integral as they describe the trajectory of a projectile. They typically appear in the form \( ax^2 + bx + c = 0 \). The equation given for the height \( h(t) \) of a projectile over time \( t \) is a quadratic equation:\[ h(t) = \frac{1}{2} a t^{2}+v_{0} t+h_{0} \]

Here, \( a \) is negative, reflecting that the projectile's upward motion is against gravity, reducing velocity over time. Solving these equations can help us determine the time at which the maximum height is reached (the vertex of the parabola), as well as the total time of flight.

• Discriminant: Helps determine the nature of roots, indicating different motion scenarios.
• Vertex Form: Useful for finding the maximum or minimum point of the projectile's path.

Specific solutions of quadratic equations allow us to pinpoint exact moments in the flight, such as when a projectile reaches zero height during its descent.

Initial Velocity

Initial velocity \( v_{0} \) is the speed at which an object begins its motion. In the context of projectile motion, especially vertically, it plays an essential role in determining how high and how long the projectile will travel.

The higher the initial velocity:

• The higher the object will go, climbing against gravity until that upward speed diminishes to zero.
• The longer the duration before the object returns to its initial level or the ground.

The initial velocity is often given in the problem, but if not, it can be found using parameters such as maximum height or time of flight, utilizing the relevant projectile motion formulas. By understanding initial velocity, we can predict and analyze the motion and behavior of projectiles effectively.

Parabolic Trajectory

A parabolic trajectory characterizes the path of a projectile under the influence of gravity, taking the shape of a parabola. This shape arises because the vertical motion of the projectile is affected by gravitational acceleration, while horizontal motion remains constant (if considered).

Key points about parabolic trajectory include:

• The highest point of the parabola is the projectile's maximum height.
• Both sides of the parabola are symmetric concerning the vertical axis.
• The time to rise to maximum height is equal to the time to fall back.

Understanding the parabolic nature helps in predicting where and when the projectile will hit the ground and is essential in fields ranging from sports to engineering. This trajectory model applies universally to objects in a gravity-driven vertical motion unless other external forces are in play.