91Ó°ÊÓ

Horizontal motion in projectile motion refers to the motion of an object in a straight path parallel to the ground. In the given exercise, the bullet fired from the rifle travels horizontally towards the target. This motion is characterized by a constant horizontal velocity since there are no horizontal forces acting on the bullet (assuming air resistance is negligible).

To determine how far the bullet travels horizontally, we use the simple distance formula:

• Distance ( \(d\)) = Horizontal Velocity ( \(v_{0x}\)) \(\times\) Time ( \(t\))

The initial horizontal velocity \( v_{0x}\) is given as \(460\, \mathrm{m/s}\). Since the horizontal velocity remains constant, the time taken to reach the target can be calculated by dividing the horizontal distance \( \mathrm{d}\) by \(v_{0x}\).

For this exercise, the horizontal distance to the target was \(45.7\, \mathrm{m}\), and the time taken was calculated to be approximately \(0.0993\, \mathrm{s}\). The constancy of horizontal motion helps predict where the projectile will land along the horizontal axis.

Unlike horizontal motion, vertical motion in a projectile follows a parabolic path and is affected by gravity. This component of motion is crucial to calculating how much a projectile drops while moving horizontally. In our scenario with the rifle, the vertical motion determines how far the bullet falls before it hits the target.

The vertical displacement or drop \( h \) of the bullet is due to the acceleration from gravity. It can be found using the formula:

• \[ h = \frac{1}{2}gt^2 \]

Where:

• \( g \) is the acceleration due to gravity \(\approx 9.81 \, \mathrm{m/s^2}\).
• \( t \) is the time of flight calculated from horizontal motion.

For this exercise, the bullet's vertical drop was found to be \(0.048\, \mathrm{m}\) or \(4.8\, \mathrm{cm}\). The vertical motion needs to be considered because this drop informs how much higher the shooter must aim above the target to ensure the projectile reaches the center once gravity acts on it.

Acceleration due to gravity is a constant force that acts on objects in free fall near the Earth's surface. Its value is approximately \(9.81\, \mathrm{m/s^2}\). This universal force affects any projectile's vertical motion, pulling it downward as it travels horizontally.

In the context of projectile motion, understanding gravity's role is crucial for calculating how high above a target a projectile, like a bullet, will drop during its flight. This includes:

• Determining the time it takes for the bullet to reach the target based on horizontal distance.
• Calculating the vertical drop that occurs within that time using the formula:

\[ h = \frac{1}{2}gt^2 \]

Gravity's influence means that even with a powerful velocity of \(460\, \mathrm{m/s}\), a bullet will still drop over the short span it travels to a target \(45.7\, \mathrm{m}\) away. This drop necessitates aiming higher to counteract gravity's effect, ensuring accuracy in hitting the intended target point.