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Magazine Chapter 1: Problem 98
Falling-Body Problems Use the formula \(h=-16 t^{2}+v_{0} t\) discussed in Example 7. How fast would a ball have to be thrown upward to reach a maximum height of 100 \(\mathrm{ft}\) ? \([\text { Hint: Use the discriminant of }\) the equation \(16 t^{2}-v_{0} t+h=0.1\)
Short Answer
Expert verified
The ball must be thrown upwards with an initial velocity of 80 ft/s.
Step by step solution
01
Identify the Problem
We need to find the initial velocity \( v_0 \) such that a ball reaches a maximum height of 100 feet when thrown vertically upward. The formula given is \( h = -16t^{2} + v_0 t \).
02
Set Up the Equation
To find when the ball reaches its maximum height, set \( h = 100 \) which implies setting up the equation \( -16t^2 + v_0 t = 100 \).
03
Rearrange the Quadratic Equation
Rearrange the equation into the standard quadratic form: \(-16t^2 + v_0 t - 100 = 0 \).
04
Apply the Discriminant
The discriminant of a quadratic equation \( ax^{2} + bx + c = 0 \) is \( b^2 - 4ac \). For a maximum height, the discriminant should be zero (indicating a unique solution for maximum height). Substitute \( a = -16 \), \( b = v_0 \), and \( c = -100 \) into the discriminant formula: \( v_0^2 - 4(-16)(-100) = 0 \).
05
Solve for Initial Velocity
Solve \( v_0^2 - 6400 = 0 \) to get \( v_0^2 = 6400 \). Then, solve for \( v_0 \) by taking the square root of both sides: \( v_0 = \sqrt{6400} \).
06
Calculate the Final Answer
Calculate \( v_0 = 80 \). This means the ball must be thrown upwards with an initial velocity of 80 ft/s to reach 100 ft.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
A quadratic equation is one of the foundational concepts in algebra, often taking the form of \( ax^2 + bx + c = 0 \). It is called quadratic because "quadra" relates to the square, and these equations always involve the variable being squared. Quadratic equations can describe a variety of real-world problems, such as the trajectory of objects. This is because their graphs are parabolas鈥攕ymmetrical, curved shapes that open upwards or downwards.
To solve an equation like \(-16t^2 + v_0 t - 100 = 0\), one can apply several methods:
To solve an equation like \(-16t^2 + v_0 t - 100 = 0\), one can apply several methods:
- Factoring: Expressing the quadratic in terms of its factors, if possible.
- Using the Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which directly incorporates the discriminant.
- Completing the Square: Transforming the equation into a perfect square trinomial.
Discriminant
The discriminant is a key component of the quadratic formula, denoted as \( b^2 - 4ac \). Its value provides significant insights into the nature of the roots of a quadratic equation. In the context of our falling-body problem, understanding the discriminant is crucial to determining the initial conditions required to achieve a specific height.
Here is why it is important:
Here is why it is important:
- A zero discriminant (\( b^2 - 4ac = 0 \)) indicates that there is exactly one real solution. This means the ball reaches the maximum height at a unique time, which is essential for finding the correct initial velocity.
- A positive discriminant means two real solutions exist, implying the ball could potentially reach the height at two different times if the context allowed.
- A negative discriminant signifies no real solutions, indicating the height is unattainable with the given parameters.
Initial Velocity
Initial velocity, often denoted as \( v_0 \), is the starting speed of a body in motion. It is a critical factor in projectile motion calculations, especially in falling-body problems like ours, where determining the initial velocity allows us to predict the trajectory of the ball accurately.
In our example, solving the equation \( v_0^2 - 6400 = 0 \) demonstrates that the initial velocity required is 80 ft/s. This velocity is crucial because:
In our example, solving the equation \( v_0^2 - 6400 = 0 \) demonstrates that the initial velocity required is 80 ft/s. This velocity is crucial because:
- It determines how high the ball will travel before gravity slows it down to a stop at the maximum height.
- It impacts how long the ball will stay in the air. A higher initial velocity increases both maximum height and total time in the trajectory.
Maximum Height
Maximum height in the context of falling-body problems refers to the highest point a projectile reaches during its motion before it begins descending. It is a point where the velocity of the ball becomes zero, because vertical speed has been completely reduced by gravity.
For our problem, the formula \( h = -16t^2 + v_0t \) helps us predict this height when rearranged to find time at which it occurs:
For our problem, the formula \( h = -16t^2 + v_0t \) helps us predict this height when rearranged to find time at which it occurs:
- At maximum height, the derivative of height with respect to time, \( v_0 - 32t = 0 \), tells us at what time the ball stops rising, allowing us to solve for \( t \).
- The calculated initial velocity and known maximum height combine to redefine the equation and verify that at this point of 100 ft, the projectile's velocity equals zero.
