Problem 47 - Precalculus with Limits

Question

Vertical Motion In Exercises 47 and \(48 ,\) an object moving vertically is at the given heights at the specified times. Find the position equation \(s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { 0 } t + s _ { 0 }\) for the object. $$ \begin{aligned} \text { At } t & = 1 \text { second } , s = 128 \text { feet } \\\ \text { At } t & = 2 \text { seconds } , s = 80 \text { feet } \\ \text { At } t & = 3 \text { seconds, } s = 0 \text { feet } \end{aligned} $$

Step-by-Step Solution

Answer

The position equation for the object is \( s = -16t^2 + 192t + 64 \)

Step 1: Define the Variables

From the problem statement, the constants are: \( a = -32 \, ft/s^2 \), \( t_1 = 1 \, s, t_2 = 2 \, s, t_3 = 3 \, s \), \( s_1 = 128 \, ft, s_2 = 80 \, ft, s_3 = 0 \, ft \). The unknowns are the initial velocity \( v _ { 0 } \) and the initial height \( s _ { 0 } \).

Step 2: Formulate the Equations

Applying the values of \( t \) and \( s \) to :\( s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { 0 } t + s _ { 0 } \), we get three equations: \( s_1 = \frac { 1 } { 2 } a t_1 ^ { 2 } + v _ { 0 } t_1 + s _ { 0 } \) \( s_2 = \frac { 1 } { 2 } a t_2 ^ { 2 } + v _ { 0 } t_2 + s _ { 0 } \) \( s_3 = \frac { 1 } { 2 } a t_3 ^ { 2 } + v _ { 0 } t_3 + s _ { 0 } \)

Step 3: Solve Simultaneoulsy

From step 2, we get three equations. We can solve these equations simultaneously to find the values of \( v _ {0} \) and \( s _ {0} \).

Step4: Solve for \( v _ {0} \) and \( s _ {0} \)

When equations are solved simultaneously, the values of \( v _ {0} \) and \( s _ {0} \) are 192 ft/s and 64 ft respectively.

Key Concepts

Position Equation

The position equation in vertical motion describes where an object will be at any given time during its flight. To break it down, the position equation is expressed as \( s = \frac{1}{2}at^2 + v_0t + s_0 \), where \( s \) represents the final position, \( a \) is the acceleration, \( t \) is the time, \( v_0 \) is the initial velocity, and \( s_0 \) is the initial position.

When dealing with an object in free fall or any motion affected by gravity, the equation can provide accurate predictions of the object's height or distance from a reference point over time. Navigating through this equation, it typically involves substituting known values and solving for the unknown parameters, which in many cases are the object's initial speed and position.

Initial Velocity

Initial velocity \( v_0 \) is a key component in describing the starting speed and direction of an object in motion. Knowing the initial velocity helps us to predict where the object will be at any subsequent point in time. When an object is thrown upwards or dropped, the initial velocity is what it starts with before gravity starts to exert its influence.

Depending on the direction of motion, initial velocity can be positive (upwards motion), negative (downwards motion), or zero (if it is dropped). It's crucial in calculations and must be accurately measured or calculated for kinematic equations to produce reliable results.

Acceleration due to Gravity

Acceleration due to gravity, which is denoted by \( g \) or \( a \) in equations, is a constant value representing the rate at which objects accelerate towards Earth when in free fall. This constant value is approximately \( -9.8 \, m/s^2 \) or \( -32 \, ft/s^2 \) depending on the units being used.

This acceleration is always directed downward toward the Earth's surface and affects every object that is in the air, whether it's dropped, thrown upwards, or projected outward. Note that we use a negative sign for this acceleration to indicate the downward direction when analyzing vertical motion mathematically.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces that cause the motion. The fundamental kinematic equations allow us to describe an object's position, velocity, and acceleration over time.

To solve kinematic problems, we often use a set of equations known as the equations of motion. These include not only the position equation but also those for finding velocity and acceleration as functions of time. By knowing a few initial conditions, such as the initial velocity, initial position, and the acceleration due to gravity, we can determine an object's state of motion at any point in its trajectory.