91Ó°ÊÓ

When discussing vertical motion, initial velocity refers to the speed of an object at the moment it starts its journey. In our exercise, we want to find out the speed of the arrow right as it left the bowstring. Initial velocity is crucial because it tells us how fast and in which direction an object will move initially.

In the exercise, we utilize the formula for vertical motion: \[ h = v_i \cdot t + \frac{1}{2} \cdot a \cdot t^2 \]This equation relates the height (\(h\)) to the initial velocity (\(v_i\)), time (\(t\)), and acceleration (\(a\), mainly the acceleration due to gravity). By rearranging this equation, we solve for the arrow's initial speed.

• The height (\(h\)) reached by the arrow is 30 meters after 2 seconds.
• Acceleration due to gravity (\(a\)) is \(-9.8 \text{ m/s}^2\) because gravity acts downwards.
• Thus, the calculation shows that the initial velocity \(v_i\) of the arrow is \(19.9 \text{ m/s}\).

Quadratic equations arise when there are squared terms involved, such as time squared in our vertical motion equation. In the exercise, determining the time for when the arrow reaches a certain height requires solving a quadratic equation. This is where the quadratic formula comes into play: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\).

For the arrow reaching 15 meters, we rewrite the vertical motion equation to have the following form:\[ -4.9t^2 + 19.9t - 15 = 0 \]Using:

• \(a = -4.9\),
• \(b = 19.9\),
• \(c = -15\),

we substitute these into the quadratic formula to find \(t\). The formula yields two potential times, but only the earliest time, \(1 \, \text{s}\), is relevant for the first occurrence of the arrow reaching that height. The other time would tell us when it descends back to that height later.

Gravity's role is a major factor when analyzing vertical motion. It is the force that pulls objects toward the Earth, giving them an acceleration denoted as \(g = 9.8 \, \text{m/s}^2\). However, since gravity acts downwards, we consider it as negative in calculations when objects are launched into the air.

This negative acceleration affects the vertical speed of objects, slowing their ascent and speeding up their descent. It helps explain why an object eventually stops rising and starts its journey back towards the ground.

• In our exercise, we used \(-9.8 \, \text{m/s}^2\) to calculate the arrow's trajectory over time.
• This value is integral to understanding the motion dynamics of any projectile in freefall conditions.

Understanding gravity's impact allows accurate prediction of how far and how fast the arrow (or any projectile) moves at any point in its vertical journey.