Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

If an object is projected vertically upward from an altitude of \(s_{0}\) feet with an initial velocity of \(v_{0} \mathrm{ft} / \mathrm{sec}\), then its distance \(s(t)\) above the ground after \(t\) seconds is $$ s(t)=-16 t^{2}+v_{0} t+s_{0} . $$ If \(s(1)=84\) and \(s(2)=116\), what are \(v_{0}\) and \(s_{0}\) ?

Step-by-Step Solution

Answer

The initial velocity \( v_0 = 80 \) ft/s and the initial altitude \( s_0 = 20 \) feet.

Step 1: Formulate the Given Conditions

We know the function for the height at time \( t \) is \( s(t) = -16t^2 + v_0 t + s_0 \). We also have two conditions, \( s(1) = 84 \) and \( s(2) = 116 \), to work with.

Step 2: Substitute into the Height Equation for t = 1

Using the equation \( s(1) = -16(1)^2 + v_0(1) + s_0 = 84 \), we simplify to get \( -16 + v_0 + s_0 = 84 \). This gives us our first equation: \( v_0 + s_0 = 100 \).

Step 3: Substitute into the Height Equation for t = 2

Now using \( t = 2 \), we have \( s(2) = -16(2)^2 + v_0(2) + s_0 = 116 \). Simplifying, \( -64 + 2v_0 + s_0 = 116 \), which leads to our second equation: \( 2v_0 + s_0 = 180 \).

Step 4: Solve the System of Equations

We now solve the system of equations: \( v_0 + s_0 = 100 \) and \( 2v_0 + s_0 = 180 \). Subtract the first equation from the second to get \( v_0 = 80 \). Substitute \( v_0 = 80 \) back into the first equation to find \( s_0 = 20 \).

Step 5: Verify the Solution

Substituting \( v_0 = 80 \) and \( s_0 = 20 \) back into the original equations confirms our solution so that both conditions \( s(1) = 84 \) and \( s(2) = 116 \) are satisfied.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air, subject to gravitational acceleration. It involves two key components of movement: a horizontal one and a vertical one. In our scenario, the focus is on vertical motion, as the object is projected straight upward and affected by gravity pulling it back down.
The fundamental concept here is the manner in which gravity influences the object's motion. Gravity acts downward, causing the object to decelerate as it ascends, stop at the peak, and accelerate downwards as it falls. The model for projectile motion in this sense can be mathematically represented by a quadratic equation in the form of:
\[ s(t) = -16 t^2 + v_0 t + s_0 \]
where:

• \(-16 t^2\) represents the gravitational pull reducing the object's height over time.
• \(v_0 t\) accounts for the initial velocity pushing the object upwards.
• \(s_0\) is the initial height from which the object is projected.

Understanding projectile motion is crucial for solving problems involving objects in motion influenced by gravity.

Initial Velocity

Initial velocity, denoted as \(v_0\), is the velocity at which an object begins its motion. In projectile motion, initial velocity plays a significant role in determining how high and far the object will travel.
In our example, the initial velocity is what pushes the object upwards against gravity. It affects how long the object remains in the air before descending back.

• It is part of the equation \(s(t) = -16 t^2 + v_0 t + s_0\).
• The linear term \(v_0 t\) signifies that with each second, the object's height increases by a factor proportional to the initial velocity.

The value of \(v_0\) is critical for determining the trajectory and maximum height reached. Identifying \(v_0\) involves using conditions or measurements taken at specific times, as seen in the problem, to solve for this unknown.

System of Equations

A system of equations consists of two or more equations with the same set of variables. These systems are solved to find the values of those variables.
In the given problem, we have a system of two equations derived from substituting times into the height equation:

• At \(t = 1\), we have: \(v_0 + s_0 = 100\).
• At \(t = 2\), we have: \(2v_0 + s_0 = 180\).

Solving this system involves:

• Finding the difference between equations to eliminate one of the variables.
• Determining one variable and substituting back to find the other.

This system effectively lets us find the two unknowns: initial velocity \(v_0\) and initial height \(s_0\). It is advantageous in practical situations where more than one piece of information allows for finding the unknowns.

Height Equation

The height equation in projectile motion problems describes how the height of an object changes over time when influenced by gravity. It is given by:
\[ s(t) = -16 t^2 + v_0 t + s_0 \]
This is a quadratic equation representing a parabolic trajectory caused by gravity's constant acceleration.

• The term \(-16 t^2\) is derived from the constant gravitational acceleration, which in imperial units, is approximately 32 ft/s"); however, the equation uses \(-16\) to factor in that the height is in feet and time is in seconds.
• \(v_0 t\) adds the upward effect of the initial velocity.
• \(s_0\) gives the starting point above the ground.

Understanding this equation helps predict the object's position at any time \(t\) and is crucial for analyzing the effects of different initial conditions on the projectile's motion.