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Answer :
Certainly! Let's address this problem step-by-step.
### Step 1: Understand the Scenario
You are given a scenario where a person jumps out of a plane at an initial height of 4000 meters. You need to determine the set of ordered pairs [tex]\((t, d)\)[/tex], where [tex]\(t\)[/tex] is the time in seconds after the jump and [tex]\(d\)[/tex] is the displacement above the ground.
### Step 2: Recognize the Influence of Gravity
When the person jumps out of the plane, they are under the influence of gravity. Assume the acceleration due to gravity ([tex]\(g\)[/tex]) as 9.81 meters per second squared (m/s²). The initial velocity ([tex]\(v_{\text{initial}}\)[/tex]) is zero as the person starts from rest.
### Step 3: Use the Kinematic Equation for Displacement
The displacement [tex]\(d\)[/tex] above the ground can be modeled by the kinematic equation for an object in free fall:
[tex]\[ h(t) = h_{\text{initial}} + v_{\text{initial}} \cdot t - \frac{1}{2} g t^2 \][/tex]
Given:
- [tex]\( h_{\text{initial}} = 4000 \)[/tex] meters (initial height)
- [tex]\( v_{\text{initial}} = 0 \)[/tex] meters per second (initial velocity)
- [tex]\( g = 9.81 \)[/tex] m/s² (acceleration due to gravity)
### Step 4: Simplify the Equation
Since the initial velocity is zero, the term [tex]\( v_{\text{initial}} \cdot t \)[/tex] becomes zero. Thus, the equation simplifies to:
[tex]\[ d = h_{\text{initial}} - \frac{1}{2} g t^2 \][/tex]
Substitute the known values:
[tex]\[ d = 4000 - \frac{1}{2} \cdot 9.81 \cdot t^2 \][/tex]
### Step 5: Final Equation for Displacement
The simplified equation for the displacement [tex]\( d \)[/tex] above the ground at any time [tex]\( t \)[/tex] is:
[tex]\[ d = 4000 - 4.905 t^2 \][/tex]
### Step 6: Form the Set of Ordered Pairs [tex]\((t, d)\)[/tex]
For each time value [tex]\( t \)[/tex], the corresponding displacement [tex]\( d \)[/tex] can be calculated using the above equation. So, the set of ordered pairs [tex]\((t, d)\)[/tex] is:
[tex]\[
(t, d) = \left( t, 4000 - 4.905 t^2 \right)
\][/tex]
### Conclusion
The complete set of ordered pairs [tex]\((t, d)\)[/tex] describes a person's displacement above the ground at any given time after jumping out of the plane. The displacement decreases as time increases due to the gravitational pull, and it follows the equation [tex]\( d = 4000 - 4.905 t^2 \)[/tex].
### Step 1: Understand the Scenario
You are given a scenario where a person jumps out of a plane at an initial height of 4000 meters. You need to determine the set of ordered pairs [tex]\((t, d)\)[/tex], where [tex]\(t\)[/tex] is the time in seconds after the jump and [tex]\(d\)[/tex] is the displacement above the ground.
### Step 2: Recognize the Influence of Gravity
When the person jumps out of the plane, they are under the influence of gravity. Assume the acceleration due to gravity ([tex]\(g\)[/tex]) as 9.81 meters per second squared (m/s²). The initial velocity ([tex]\(v_{\text{initial}}\)[/tex]) is zero as the person starts from rest.
### Step 3: Use the Kinematic Equation for Displacement
The displacement [tex]\(d\)[/tex] above the ground can be modeled by the kinematic equation for an object in free fall:
[tex]\[ h(t) = h_{\text{initial}} + v_{\text{initial}} \cdot t - \frac{1}{2} g t^2 \][/tex]
Given:
- [tex]\( h_{\text{initial}} = 4000 \)[/tex] meters (initial height)
- [tex]\( v_{\text{initial}} = 0 \)[/tex] meters per second (initial velocity)
- [tex]\( g = 9.81 \)[/tex] m/s² (acceleration due to gravity)
### Step 4: Simplify the Equation
Since the initial velocity is zero, the term [tex]\( v_{\text{initial}} \cdot t \)[/tex] becomes zero. Thus, the equation simplifies to:
[tex]\[ d = h_{\text{initial}} - \frac{1}{2} g t^2 \][/tex]
Substitute the known values:
[tex]\[ d = 4000 - \frac{1}{2} \cdot 9.81 \cdot t^2 \][/tex]
### Step 5: Final Equation for Displacement
The simplified equation for the displacement [tex]\( d \)[/tex] above the ground at any time [tex]\( t \)[/tex] is:
[tex]\[ d = 4000 - 4.905 t^2 \][/tex]
### Step 6: Form the Set of Ordered Pairs [tex]\((t, d)\)[/tex]
For each time value [tex]\( t \)[/tex], the corresponding displacement [tex]\( d \)[/tex] can be calculated using the above equation. So, the set of ordered pairs [tex]\((t, d)\)[/tex] is:
[tex]\[
(t, d) = \left( t, 4000 - 4.905 t^2 \right)
\][/tex]
### Conclusion
The complete set of ordered pairs [tex]\((t, d)\)[/tex] describes a person's displacement above the ground at any given time after jumping out of the plane. The displacement decreases as time increases due to the gravitational pull, and it follows the equation [tex]\( d = 4000 - 4.905 t^2 \)[/tex].
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