Question

In a volleyball game, a player on one team spikes the ball over the net when the ball is 10 feet above the court. The spike drives the ball downward with an initial velocity of 55 feet per second. How much time does the opposing team have to return the ball before it touches the court?

Short answer

The opposing team has approximately 1.54 seconds to return the ball before it touches the court.

Step-by-step solution

Setup the given information

Insert the given information into the motion model. We understand that the ball is 10 feet above so \(h = 10\), it is traveling downwards at 55 feet per second so \(v = -55\) (negative because the direction is downwards), and \(g = 32.2\) feet per second squared (standard acceleration due to gravity). Set \(h = 0\) since we want to know when the ball hits the ground.

Insert the values to the equation

Substitute the known values into the equation to get: \(0 = -55t - 0.5 * 32.2 * t^2 + 10\). Rearrange the equation to \(0.5 * 32.2 * t^2 + 55t - 10 = 0\)

Solve quadratic equation

The equation obtained is a quadratic equation, in the form \(at^2 + bt + c = 0\), where \(a = 0.5 * 32.2\), \(b = 55\), and \(c = -10\). The quadratic equation can be solved for \(t\) by applying the quadratic formula \(t = [-b \pm \sqrt{(b^2 - 4ac)}]/(2a)\). Here we consider only the positive root, since time can not be negative.

Calculate

Calculate \(t\) which gives two possible answers, always take the positive answer as time cannot be negative. After calculating, find that the time is roughly 1.54 seconds

Key concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is pivotal in numerous areas of mathematics and physics, including motion problems. A quadratic equation is generally of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable we want to solve for.

In the context of the volleyball game, the motion of the ball after being spiked can be described using such an equation. Solving the quadratic equation is essentially finding the values of \( t \) (in our case, time) that make the equation true. To do this, we most often use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The \( \pm \) sign indicates that there are usually two solutions, a positive and a negative. However, since we are dealing with time, we only consider the positive value as a valid answer because time cannot be negative.

In our problem, after arranging the given information into the standard form, we replace \( a \) with \( 0.5 \times 32.2 \), \( b \) with the initial velocity \( v = -55 \), and \( c \) with \( -10 \) (the height from which the ball was spiked). Solving this gives us the time the opposing team has to react to the ball.

Initial Velocity

When considering motion, especially in the context of projectile motion like a volleyball being spiked, the initial velocity is a crucial factor. The initial velocity is the speed and direction in which an object starts its journey.

In our step-by-step exercise, the initial velocity is given as 55 feet per second downward, which is why it's represented as \( -55 \) when we insert it into the motion equation. The negative sign here is important as it indicates the direction of motion: since the ball is moving downward, the initial velocity is considered negative in the upward direction of the standard coordinate system used for motion problems.

This initial velocity greatly determines the trajectory and the impact time, which is the time it takes for the ball to reach the court. The accurate determination of initial velocity is essential for predicting the outcome of the ball's motion and thus determining how much time the opposing team has to react.

Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The motion of the spiked volleyball follows a parabolic trajectory under the influence of gravity, and this motion can be analyzed using quadratic equations.

The equation that describes projectile motion in our exercise includes the effects of gravity, which is why the acceleration due to gravity \( g = 32.2 \text{ feet per second squared} \) is a part of the quadratic equation. The term \( -0.5 \times g \times t^2 \) in the equation accounts for the distance the ball falls under gravity over time \( t \) after being spiked with an initial velocity.

Importance of Accurate Calculations

Accurate calculations in projectile motion are critical for players to know how much time they have to position themselves and react to the ball. Understanding the underlying physics can significantly improve how players anticipate the ball's path and the required response time.

Real-world Applications

Moreover, the principles of projectile motion apply not only to sports but also to engineering, ballistic trajectories, and even space travel, making it a universal concept in physics.