Problem 12 - Algebra 2

Question

BASKETBALL The maximum height \(H\) in feet that a basketball reaches after being shot is given by the formula \(H=\frac{V_{0}^{2}(\sin \theta)^{2}}{64},\) where \(V_{0}\) represents the formula \(H=\frac{V_{0}^{2}(\sin \theta)^{2}}{64},\) where \(V_{0}\) represents the initial velocity and \(\theta\) represents the degree measure of the angle that the path of the basketball makes with the ground. Find the maximum height reached by a ball shot with an initial velocity of 30 feet per second at an angle of \(70^{\circ} .\)

Step-by-Step Solution

Answer

The maximum height is approximately 12.42 feet.

Step 1: Substitute the Given Values

In the formula \(H = \frac{V_0^2 (\sin \theta)^2}{64}\), substitute \(V_0 = 30\) and \(\theta = 70^{\circ}\). This gives us \(H = \frac{30^2 (\sin 70^{\circ})^2}{64}\).

Step 2: Calculate \(30^2\)

Square the initial velocity: \(30^2 = 900\). This simplifies our expression to \(H = \frac{900 (\sin 70^{\circ})^2}{64}\).

Step 3: Find \(\sin 70^{\circ}\)

Calculate \(\sin 70^{\circ}\) using a calculator. \(\sin 70^{\circ} \approx 0.9397\).

Step 4: Square \(\sin 70^{\circ}\)

Calculate \((\sin 70^{\circ})^2 = (0.9397)^2 \approx 0.8836\).

Step 5: Calculate Maximum Height

Now substitute \((\sin 70^{\circ})^2 \approx 0.8836\) into the height equation: \(H = \frac{900 \times 0.8836}{64}\).

Step 6: Compute the Final Result

Perform the multiplication and division: \(H = \frac{795.24}{64} \approx 12.42\). Thus, the maximum height is approximately 12.42 feet.

Key Concepts

Projectile Motion

Projectile motion is an interesting topic in physics that involves any object thrown into space upon which the only acting force is gravity. In the case of shooting a basketball, the ball follows a parabolic trajectory, which is a hallmark of projectile motion.

The maximum height a basketball can reach depends on several factors:

• Initial velocity with which the ball is launched.
• The angle at which the basketball is shot.
• The gravitational force acting downwards on the basketball.

Understanding how these elements interact helps players optimize their shots for better performance, whether they're shooting hoops on the court or solving equations in their math homework.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, play a crucial role in calculating projectile motion. In this scenario, we use the sine function to determine how high the basketball will go when shot at a particular angle.

The sine function is especially useful because it relates the angle of the shot (angle of projection) to the opposite side's length in a right triangle which, in a projectile problem, corresponds to the vertical component of the initial velocity. By using \(\sin \theta\), we can easily find the extent to which the initial velocity influences the vertical rise of the basketball.

In the formula provided, \(\sin \theta\) is squared because we look for the vertical height at the top of the projectile's path, which requires balancing out the initial upward push with the downward pull of gravity.

Initial Velocity

The initial velocity, denoted by \(V_0\), is a determining factor in projectile motion and specifically in calculating the maximum height the basketball can reach. The initial velocity refers to the speed at which the ball is propelled into the air right after it leaves the player's hands.

When you see terms like \(V_0^2\) in equations, it's due to the quadratic nature of kinetic energy and related motion. By squaring the initial velocity, the equation takes into account both the speed and the additional influence it has on the ball's trajectory in all directions.

• Higher initial velocity means higher potential maximum height.
• If the initial velocity is zero, the ball won't rise at all.

Knowing how to manipulate initial velocity can help players achieve accurate shots in games.

Angle of Projection

The angle of projection, denoted by \(\theta\), is the angle at which the basketball is launched relative to the ground. This angle influences how high and how far the ball can travel.

An optimal angle often leads to the best combination of height and range, which can be crucial for scoring points in a game. For example, \(\theta = 45^\circ\) is often cited as the best angle for maximizing distance in theory, providing an ideal balance between horizontal and vertical velocities.

• Adjusting the angle affects the ball's trajectory significantly.
• A steeper angle, close to \(90^\circ\), focuses on height over distance.
• A shallower angle, near \(0^\circ\), focuses on distance over height.

Understanding the angle of projection can greatly enhance a basketball player's shot precision, mimicking a real-life physics calculation.