Question

A projectile is fired straight upward from ground level with an initial velocity of 160 feet per second. During what time period will its height be less than 384 feet?

Short answer

The time period during which its height will be less than 384 feet is between 4 seconds and 6 seconds.

Step-by-step solution

Identify Knowns and Unknowns

The initial velocity \(v_0\) is given as 160 feet per second, the acceleration \(a\) due to gravity is -32 feet per second squared as it is acting downwards which is conventionally chosen as the negative direction, and the maximum height that we need to find the time for, \(h\), is 384 feet. We need to find the time \(t\) it takes for the projectile to reach this height.

Use the equation of motion

We use the equation of motion \(h = v_0 t - 0.5 a t^2\), where \(h\) is the height, \(v_0\) is the initial velocity, \(t\) is time and \(a\) is acceleration.

Insert values and solve for time

Substituting the given values into the equation, we get: \(384 = 160t - 0.5*32t^2\). This rearranges to: \(0.5*32t^2 - 160t + 384 = 0\). Simplify further, it becomes: \(16t^2 - 160t + 384 = 0\). Dividing throughout by 16 to simplify, we get: \(t^2 - 10t + 24 = 0\). This quadratic equation can be factored to: \((t-4)(t-6)=0\). Hence \(t=4\) and \(t=6\) seconds are the times at which the height is exactly 384 feet. In between these times, the height is less than 384 feet.

Key concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \). In these equations, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable of interest. Quadratics are fundamental in calculating projectile motion paths because they naturally model the arcs formed by projectiles in motion.

When solving a quadratic equation, we aim to find the values of \( x \) that make the equation true. Techniques for solving these include factoring, using the quadratic formula, or completing the square. In our projectile problem, we factored the quadratic equation \( t^2 - 10t + 24 = 0 \) to find the times at which the projectile reaches specific heights.

• Factoring simplifies the equation if possible, as seen with \((t-4)(t-6)=0\).
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) is a universal method.

Understanding how to manipulate and solve these equations is key to analyzing projectile motion.

Initial Velocity

The initial velocity in a projectile motion problem is crucial because it determines the projectile's initial kinetic energy and how far and high it will travel. In our exercise, the initial velocity \( v_0 \) is 160 feet per second. Initial velocity is usually denoted as \( v_0 \), where the subscript zero indicates the starting point.

The importance of knowing the initial velocity lies in how it affects the motion:

• Higher initial velocities result in farther travel distances.
• It influences the time a projectile is in motion, affecting its time of flight.
• The direction also matters. Here, the velocity is upward.

In projectile motion equations, the initial velocity forms part of the linear component, often expressed as \( v_0 t \), which describes how the projectile's airborne time is directly related to its speed at launch.

Acceleration due to Gravity

Acceleration due to gravity is a constant force that affects the motion of all objects flying through the air. It is denoted as \( g \) and usually acts downward on a projectile. On Earth, this acceleration is approximately \( -32 \) feet per second squared, which is why you see \'-32\' in the calculations. This negative sign indicates that gravity acts in the opposite direction to the projectile's initial upward motion.

The key effects of gravitational acceleration include:

• Slowing the projectile's ascent as it moves upwards.
• Accelerating its descent back toward the ground.
• Causing the characteristic parabolic path of projectile motion.

In the quadratic equation \( h = v_0 t - 0.5 a t^2 \), the \( -0.5 a t^2 \) term accounts for the reduction in height due to gravity over time. Knowing gravity's effects allows us to accurately predict how long and how far a projectile can travel.

Time of Flight

Time of flight is the total time that a projectile is in motion from the moment it is launched until it returns to its release point or ground level. Calculating the time of flight helps determine key events, like when the projectile reaches certain heights. In the exercise solution, we determined that there are two times, \( t=4 \) seconds and \( t=6 \) seconds, when the projectile is exactly 384 feet high.

Between these times, the projectile's height is less than 384 feet. Understanding the time of flight in projectile motion helps predict whether the trajectory will meet certain criteria. Here are some insights regarding time of flight:

• Longer initial velocities extend the time of flight.
• Gravity ultimately limits how long the projectile remains airborne.
• The total time of flight can be symmetrically split; from launch to peak, and peak to return.

An accurate calculation of time of flight helps anticipate and understand the projectile's full journey, essential for a range of real-world applications.