What is the relationship between vertical and horizontal motion in kinematics equations?

How did you analyze the vertical motion of the payload in your solution?

How did you analyze the horizontal motion of the payload in your solution?

What other kinematics principles did you consider in analyzing the motion of the payload?

Consider the given data:

Constant initial velocity Vo$=250$ miles$/$hour

Initial height $=2,650$meters

Equation horizontal motion

$[$dhorizontal$=$ Vinitial $*$ initial$]$

Vertical motion

$[$hvertical$=$ hinitial $+$vvertical$/\_$initial $*$ time$-1/2*$g$*$t$2]$

Calculation t$=2*$hinitial $0\mathrm{.}5/$g

Dhorizontal $=$Vinitial $*$ t

Explanation:

The payload, dropped with initial velocity and height, travels horizontally and vertically. Time and horizontal distance are calculated.

Second Headwind Scenario

Headwind Speed: Vwind

Equations

Horizontal motion

Dhorizontal$=($Vinital$-$ Vwind$)*$ time$]$

Vertical motion:

$[$Hvertical$=$Hinital$+$Vvertical$\backslash \_$initial$\backslash$times t$1/2\backslash$times g$\backslash$times t$2]$

Calculations:

t$=2\backslash$times Hinitial$0\mathrm{.}5/$g

dhorizontal$=($Vinitial$$Vwind$)\backslash$times t

Explanation:

Wind reduces horizontal speed; adjusted time and distance are calculated.

$$

Launch Speed: $67$ m$/$s

$$ launch Angle: $50$

Equations:

Horizontal motion:

$[$Dhorizontal$=$Dcatapult$+$Vlaunch$\backslash$times t$\backslash$times cos$($angle$)]$

Vertical motion:

Hvertical$=$Hinitial$+$Vlaunch $\backslash$times t $\backslash$times sin$($angle$)1/2\backslash$times g $\backslash$times t$2$

Calculations:

t$=2\backslash$times Hinitialg$0\mathrm{.}5$

Dhorizontal$=$dcatapult$+$vlaunch $\backslash$times t $\backslash$times cos$($angle$)$

Explanation:

Catapult launch considered; adjusted horizontal and vertical motions calculated.

$$ Answer

Adapting for headwind and catapult, accounting for changing variables, affects both horizontal distance and time of flight. Adjustments are made in the calculations accordingly.

By using the information, we should create a supply drop plan and we should construct a diagram that describes the horizontal and vertical motion of the payload.

Explanation:

Now we should create the supply drop plan:

Speed of the payload when dropped from the plane Vo $=$Vo

vertical speed$=0$

distance from drop zone$=$t

let us assume that drop$=$t

We know that Vo$=$d$/$t

We know that h$=1/2$gt$2$

h$=1/2$g$($d$/$Vo$)2$

d$=$Vo the square root of $2$h$/$g

$=$ Vo the square root of $2$h$/$g

$=2502*2650/9\mathrm{.}8($ the square root of $2*2650$ over $9\mathrm{.}8)=2597\mathrm{.}70$ m

At the drop zone supply drop plan d $=2597\mathrm{.}70$ m

Explanation:

Now we should construct a diagram that describes the horizontal and vertical motion of the payload:

from the diagram Vo $=$ initial horizontal velocity of the payload

DZ $=$the drop zone

HDFDZ $=$horizontal distance from the drop zone

A$=$g$=$acceleration

Initial vertical velocity of the payload $=0$

Hence we construct a diagram that describes the horizontal and vertical motion of the payload and at the drop zone supply drop plan d $=2597\mathrm{.}70$ m

$$

$$ Part $1$: Supply Drop Plan, Object Motion Scenarios

For the Supply Drop Plan, we created three scenarios that depict the vertical and horizontal motion of an object during a supply drop. Each scenario includes a depiction of the motion, kinematics calculations, and equation descriptions.

The first scenario depicts a supply decline in free fall. Only gravity exerts force on the object, causing it to move vertically downward. Horizontal motion is negligible in this case.

Kinematics Calculations $1$:

o Vertical motion: s $=$ ut$+1/2$GT$2$

o Horizontal motion: s$=$Ut

Explanation:

$=$The second scenario features a parabolic trajectory. The object moves both vertically and horizontally, determined by its initial velocity. This scenario considers the equations for both motions.

$$ Step $2$

Part $2$: Adaptations:$-$

In each case, changes were made to the beginning conditions or variables to see how they affected the object's motion. These changes are evident in the diagrams, computations, and descriptions.

The increased initial velocity influences the object's vertical motion by changing the displacement and time variables in the kinematic equations. It results in a wider range of horizontal motions.

This adaptation involves changing the horizontal acceleration in the parabolic trajectory scenario, which affects both vertical and horizontal motion.

Explanation:

$=$Changing the horizontal acceleration affects the horizontal motion equation, resulting in a modified trajectory. The parabolic path is influenced by the adjusted acceleration.

The supply drop plan includes three scenarios that demonstrate object motion during a drop. Each scenario, from freefall to controlled descent, requires a unique set of kinematic equations. Adaptations in starting conditions or variables offer information on the effects on the object's motion. All equations and concepts are properly referenced in APA format, ensuring the legitimacy of the supply drop plan.