Problem 34 - Precalculus: Functions and Graphs

Question

If an object is projected vertically upward from an altitude of \(s_{0}\) feet with an initial velocity of \(v_{0}\) ft \(/\) 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 \(v_0\) is 80 ft/s, and the initial altitude \(s_0\) is 20 feet.

Step 1: Substitute the First Condition

We are given that when \(t = 1\), \(s(1) = 84\). Substitute \(t = 1\) into the equation:\[s(1) = -16(1)^2 + v_0(1) + s_0 = 84\]This simplifies to:\[-16 + v_0 + s_0 = 84\]Rearrange to get the first equation:\[v_0 + s_0 = 100\]

Step 2: Substitute the Second Condition

Next, we are given that when \(t = 2\), \(s(2) = 116\). Substitute \(t = 2\) into the equation:\[s(2) = -16(2)^2 + v_0(2) + s_0 = 116\]This simplifies to:\[-64 + 2v_0 + s_0 = 116\]Rearrange to get the second equation:\[2v_0 + s_0 = 180\]

Step 3: Solve the System of Equations

We now have a system of two equations:1. \(v_0 + s_0 = 100\)2. \(2v_0 + s_0 = 180\)Subtract the first equation from the second:\[(2v_0 + s_0) - (v_0 + s_0) = 180 - 100\]Simplify to find:\[v_0 = 80\]

Step 4: Find the Value of s_0

Use the value of \(v_0 = 80\) in the first equation: \[v_0 + s_0 = 100\]Substitute \(v_0 = 80\):\[80 + s_0 = 100\]Solve for \(s_0\):\[s_0 = 20\]

Key Concepts

Kinematic Equations

Kinematic equations form the backbone of understanding projectile motion in physics. They describe the motion of objects based on certain known variables such as time, initial velocity, and displacement. The main purpose of kinematic equations is to relate these variables without considering the forces that cause the motion.

• The generalized equation that applies here is: \[s(t) = -16 t^2 + v_0 t + s_0\]

This specific equation accounts for the vertical motion of an object under the influence of gravity, with the term \(-16 t^2\) representing the effect of gravity on the projectile. The negative sign signifies that gravity is acting downward.These equations are essential because they help us predict various aspects of an object's motion once it's in the air. They give a clear connection between variables, allowing us to solve for unknowns like initial velocity and starting altitude.

Vertical Motion

Vertical motion is a type of projectile motion where the object moves along a vertical path. Unlike horizontal motion, vertical motion is affected by gravitational pull, which acts constantly downward.
In the equation \[s(t) = -16 t^2 + v_0 t + s_0\]these elements tell us:

• -16 represents the influence of gravity, specifically \-32 ft/s^2\. It is halved in the equation because it accounts for accelerated motion over time.
• The \(v_0 t\) term represents how the initial velocity influences the journey upwards.

This type of motion can help predict how high or how fast an object will move upwards before gravitational pull brings it back down. Understanding vertical motion, particularly how time and gravity interact, allows for more precise predictions about an object's trajectory.

Initial Velocity

Initial velocity, represented by \(v_0\), is the speed at which an object starts its motion. It is crucial for determining how quickly and to what extent an object will ascend in vertical motion.
In the given problem, we use initial velocity to find out other variables in the kinematic equation. Knowing the value of \(v_0\) helps us determine the object's trajectory because:

• It influences the time taken to reach the peak of the motion.
• It contributes to how altitude changes over time.

If the initial velocity is higher, the object will naturally reach a higher point before it starts descending.In the exercise, we calculated \(v_0\) to be 80 ft/s, meaning each second of motion adds 80 feet to the altitude, minus the effect of gravity. Understanding initial velocity means understanding key dynamics of projectile motion.

Altitude

Altitude in the context of this problem represents the height of the object above the ground at any given time. The variable \(s_0\) is the initial altitude, where the object starts its journey.
This determines the starting position when the timer begins, essentially the zero-point of calculations. The altitude helps us understand the entire path and height covered by the object from the point it's been projected.
The variables combined can give us a complete snapshot:

• The higher the initial altitude, the longer the object can remain in the air because it has a head start before gravity pulls it back.
• In our exercise, \(s_0\) was found to be 20 feet, which means the object begins 20 feet above the ground.

Understanding altitude serves as a reference point for the object's trajectory and vertical displacement as time progresses.