Problem 96 - Algebra and Trigonometry Real Mathematics, Real People

Question

A projectile is fired straight upward from ground level with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet?

Step-by-Step Solution

Answer

(a) The projectile will be back at ground level in 8 seconds. (b) The height will be less than 128 feet after approximately 6 seconds or in approximately 2 seconds of the onset of descent.

Step 1: Find time of flight

First, set \(v\) to zero to find when the projectile will reach its apex and start its descent. Use the equation \(v = u + gt\). In this instance, \(v = 0 ft/sec\); \(u = 128 ft/sec\) and \(g = -32 ft/sec^2\) which yields \(t = \frac{-u}{g}\) after a little rearrangement. Substituting known values, solve for \(t\).

Step 2: Find total time back to ground

The total time for the projectile to return back is twice the time to reach maximum height. Hence multiply the time obtained from step 1 by 2 to get the total time.

Step 3: Find time when height is less than 128 feet

Use \(s = ut + \frac{1}{2}gt^2\) to solve for \(t\). Set \(s = 128 ft\), \(u = 128 ft/sec\), \(g = -32 ft/sec^2\) to get a quadratic equation in \(t\). After solving the quadratic equation, you get two roots. The smaller root is when the projectile is going up while the larger root is when it's coming down. Hence use the larger root.

Key Concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. Understanding kinematics is crucial when analyzing projectile motion because it helps us describe how an object moves from one point to another.
In the context of projectile motion, the basic kinematic equations are used to determine parameters such as velocity, acceleration, displacement, and time.

• The initial velocity, given as 128 ft/sec, acts as the starting speed of the projectile when launched straight upward.
• The acceleration due to gravity, denoted by \( g \), is \(-32 \) ft/sec². This value is constant and represents the gravitational pull that slows down then accelerates the projectile as it falls back down to the ground.
• These known values are crucial inputs into kinematic equations that will calculate the time it takes for the projectile to reach its maximum height or return back to ground level.

By mastering kinematics, you can easily predict and explain various aspects of the projectile's behavior, and these foundational principles apply to any motion involving acceleration and initial velocity.

Quadratic equations

When reviewing projectile motion, quadratic equations often arise when using kinematic formulas to solve for displacement or height over time. Let’s explore this in more detail.
Quadratic equations take the form \( ax^2 + bx + c = 0 \). In projectile motion, you often encounter this form when dealing with the equation for displacement, given by \( s = ut + \frac{1}{2}gt^2 \). Here's a breakdown of the components:

• The term \( ut \) represents the initial upward movement of the projectile.
• The term \( \frac{1}{2}gt^2 \) is related to the influence of gravity on the projectile over time.
• The solution to the quadratic equation is found using either the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) or by factoring, when possible.

For example, setting the height \( s \) to 128 ft creates a quadratic equation in terms of \( t \). The roots of the equation reveal when the height is precisely 128 feet. Understanding how to apply and solve these equations is key to answering many projectile movement questions.

Physics of motion

The physics of motion within the context of projectile problems involves analyzing the various stages an object undergoes during its path. Projectile motion refers to an object thrown into the air subject to only the acceleration of gravity.
The study of the physics of projectile motion includes:

• Determining the time the projectile takes to reach its peak (where its vertical velocity is zero).
• Calculating the total flight time by assessing the symmetry in the upward and downward paths. For instance, in the given exercise, once the maximum height is reached, the time taken to come back down is the same, resulting in a total flight time double the ascent time.
• Understanding when and how the height of the projectile becomes lesser than a given value is also crucial. Employing kinematic equations sheds light on these dynamics by solving quadratic equations lending insights into different phases of motion.

Apprehending the physics behind motion ensures you can intuitively analyze projectile scenarios, predict outcomes, and comprehend real-life applications of such foundational physics.