Problem 30 - Algebra 2

Question

For Exercises 30 and \(31,\) use the formula \(h(t)=v_{0} t-16 t^{2}\) where \(h(t)\) is the height of an object in feet, \(v_{0}\) is the object's initial velocity in feet per second, and \(t\) is the time in seconds. TENNIS A tennis ball is hit upward with a velocity of 48 feet per second. Ignoring the height of the tennis player, how long does it take for the ball to fall to the ground?

Step-by-Step Solution

Answer

The ball takes 3 seconds to fall back to the ground.

Step 1: Understand the Problem

We need to determine the time it takes for a tennis ball, hit upwards with an initial velocity of 48 feet per second, to return to the ground. In this case, the height \( h(t) \) becomes zero when the ball touches the ground.

Step 2: Set Up the Equation

Using the given formula \( h(t) = v_{0} t - 16t^{2} \), substitute \( v_{0} = 48 \) (initial velocity) and \( h(t) = 0 \) (height at the ground level), giving: \[ 0 = 48t - 16t^{2} \].

Step 3: Factor the Equation

To solve \( 0 = 48t - 16t^{2} \), factor out \( t \): \[ t(48 - 16t) = 0 \].

Step 4: Solve for Time \( t \)

This equation gives two solutions: \( t = 0 \) (initial time when the ball is hit) and \( 48 - 16t = 0 \). Solve \( 48 - 16t = 0 \) for \( t \): \[ 16t = 48 \] which simplifies to \( t = 3 \).

Step 5: Conclusion

The ball takes 3 seconds to reach the ground, as \( t = 0 \) is the moment the ball was initially hit, meaning \( t = 3 \) is the time it takes to come back down.

Key Concepts

Quadratic Equations

Quadratic equations play a key role in solving problems related to projectile motion. In our exercise, the equation \( h(t) = v_{0} t - 16t^{2} \) is used to model the height of a tennis ball as a function of time. Here's why quadratic equations are crucial:

• They help us understand how an object moves in a curve, known as a parabola, when subject to constant acceleration, such as gravity.
• The general form of a quadratic equation is \( ax^{2} + bx + c = 0 \), where \( a, b, \) and \( c \) are constants.
• These equations often appear in physics to relate different physical quantities, like velocity, time, and height.

By setting the equation equal to zero, we can find points of intersection, shedding light on when the ball hits the ground. When we factor this quadratic equation, it gives us potential solutions for time \( t \), revealing not just when the object was initially thrown, but also when it returns to its starting elevation.

Initial Velocity

Initial velocity is the speed at which an object starts its motion. In projectile motion, it significantly influences how far and how long something will travel. For the tennis ball in our exercise:

• The initial velocity \( v_0 \) is given as \( 48 \) feet per second.
• This value was plugged into the equation \( h(t) = v_{0} t - 16t^{2} \).

Impact of Initial Velocity

Higher initial velocity means that the tennis ball will travel higher and stay in the air longer. Initial velocity also determines the peak height the object can reach before descending due to gravity. By analyzing the initial velocity, students can predict and calculate outcomes such as time of flight and maximum height.

Time of Flight

Time of flight refers to the duration an object will be in the air. It's a vital concept when studying projectile motion, providing insights into the object's complete journey. For our tennis ball example:

• The time of flight is the total time for the ball to rise and then fall back to the ground.
• Using the formula, we solved for time \( t \) by substituting values, and found that it takes 3 seconds for the ball to complete its journey to the ground.

By calculating the time of flight, you can deduce key information about the motion of objects under gravity's influence. It is the sum of time taken to reach the peak height and the time it takes to descend back to the original height.

Free Fall

Free fall describes the motion of an object under the influence of gravity alone, neglecting other forces like air resistance. In our calculation:

• The term \( -16t^{2} \) in the equation accounts for gravitational acceleration, approximated as \( 32 \) feet per second squared, thus \( -16 \) since it's half for the equation derivation.
• When a ball is in free fall, the only force acting on it is gravity, which constantly accelerates it downwards.

The concept of free fall allows us to predict how quickly the speed of an object will increase as it falls back to the ground, enabling students to answer questions about speed and distance during its descent.