Problem 26 - Algebra 2 and Trigonometry

Question

Gravity on the moon is about one-sixth of gravity on Earth. An astronaut standing on a tower 20 feet above the moon's surface throws a ball upward with a velocity of 30 feet per second. The height of the ball at any time \(t\) (in seconds) is \(\mathrm{h}(t)=-2.67 t^{2}+30 t+20\) To the nearest tenth of a second, how long will it take for the ball to hit the ground?

Step-by-Step Solution

Answer

The ball will hit the ground at approximately 11.6 seconds.

Step 1: Identify the problem

We need to find the time at which the ball hits the ground. This means we need to solve for the time when the height of the ball, \(h(t)\), is zero.

Step 2: Set the equation to zero

To find when the ball hits the ground, set the height equation \( h(t) = -2.67t^2 + 30t + 20 \) equal to zero: \( -2.67t^2 + 30t + 20 = 0 \).

Step 3: Apply the quadratic formula

To solve the quadratic equation \( -2.67t^2 + 30t + 20 = 0 \), use the quadratic formula: \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -2.67 \), \( b = 30 \), and \( c = 20 \).

Step 4: Calculate the discriminant

Compute the discriminant: \( b^2 - 4ac = 30^2 - 4(-2.67)(20) \). This calculation gives the value under the square root.

Step 5: Evaluate the quadratic formula

Substitute the values: \( t = \frac{-30 \pm \sqrt{900 + 213.6}}{-5.34} \). Calculate the roots.

Step 6: Determine the positive root

Since time cannot be negative, use only the positive root from the quadratic formula. Compute and round to the nearest tenth.

Key Concepts

Gravity

Gravity is the force that pulls objects towards each other. On Earth, we experience this as the force that keeps us grounded. Gravity on the Moon is weaker—only about one-sixth of Earth's gravity. This difference significantly affects how objects behave when thrown on the Moon compared to Earth. For the given problem, the gravity on the Moon is incorporated into the quadratic equation as a coefficient of the term involving time squared. This coefficient determines how fast the velocity of an object changes as it moves upward or downward. When gravity is weaker, as on the Moon, objects fall more slowly, giving them a longer hang-time in the air. This is why the height equation for the Moon differs from what it would be on Earth.Recognizing this difference helps understand the problem context and how gravitational forces affect projectile motion.

Projectile Motion

Projectile motion involves any object moving through space influenced only by gravity. When the astronaut throws the ball upwards, it follows a curved path due to gravity's pull. This motion is described by specific equations, and in the Moon's case, it is characterized by a quadratic equation.

• Vertical motion is influenced by gravity, reducing the ball’s upward speed until it eventually turns back toward the surface.
• The shape of the path—the parabola—is defined by the equation of motion: \[ h(t) = -2.67t^2 + 30t + 20 \].
• Because of the Moon's reduced gravity, the ball stays in the air longer compared to Earth.

Studying projectile motion helps in predicting an object's path, which is crucial for applications like space exploration and flight.

Discriminant

The discriminant is a component of the quadratic formula, \( b^2 - 4ac \). It is essential because it tells us about the nature of the roots of the quadratic equation:

• If the discriminant is positive, the equation has two distinct real roots.
• If it's zero, there is exactly one real root, meaning the object reaches maximum height without touching the ground afterwards.
• If negative, the roots are complex, indicating that the ball theoretically never hits the ground (impossible unless dealing with hypothetical scenarios).

In the given problem, the discriminant helps in finding when the ball will land back on the Moon’s surface by providing insight into the solutions of the height equation. By computing it as part of the quadratic formula method, you establish whether real time solutions exist for when the ball hits the ground.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations and is applicable to our situation of finding when the ball hits the Moon's surface. This formula is: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula offers:

• A methodical way to find the roots of any quadratic equation, regardless of their complexity.
• The quadratic formula consists of coefficients \(a\), \(b\), and \(c\), which in our problem are defined by the equation parameters \(-2.67\), \(30\), and \(20\).
• A built-in way to evaluate the discriminant, determining the nature and number of solutions (roots) the equation has.

The correct application fosters accurate computation of roots and is crucial in determining the time taken for the ball's descent back to the Moon's crust. Understanding the quadratic formula, therefore, plays a critical role in resolving height equations derived from projectile motion problems.