91Ó°ÊÓ

Freefall acceleration is a fundamental concept in the study of projectile motion. It describes the acceleration of an object under the influence of gravity, without any other forces acting on it. On Earth, this acceleration is approximately \(9.8 \text{ m/s}^2\). This value, often represented as \(g\), remains constant irrespective of the weight or shape of the falling object, provided air resistance is negligible.
In situations involving projectile motion, like in our exercise, the downward force of gravity acts to accelerate the object, causing it to follow a curved path. This is particularly evident when the projectile is fired horizontally. Because the only force acting vertically on the projectile is gravity, we can consider its vertical motion independently from any horizontal movement. This allows us to use the formula for vertical motion under constant acceleration: \[y = 0.5gt^2\] where \(y\) is the vertical displacement, \(g\) is the acceleration due to gravity, and \(t\) is the time. This equation helps determine how far an object has fallen vertically over time due to gravity alone. Understanding this concept is key in working with projectile problems.

Projectile motion is intriguing because it combines two independent movements: horizontal and vertical. When you break these movements down, you see that they do not affect each other. That's the beauty of projectile motion!### Horizontal MotionIn horizontal motion, where the velocity is constant because there are no forces acting horizontally (ignoring air resistance), we can use the formula:- \(x = v_{xi}t\)Here, \(x\) is the horizontal distance, \(v_{xi}\) is the initial horizontal velocity (also known as muzzle velocity), and \(t\) is the time. Since the initial vertical velocity is zero in our problem (the projectile is shot horizontally), this relates directly to the time the projectile is in motion.### Vertical MotionConversely, vertical motion is affected by gravity. The vertical velocity starts at zero but increases as the object falls. This is described by the equation:- \(y = 0.5gt^2\)Thus, the projectile's position can be determined by calculating how far it fell vertically due to gravity. These two sets of motion equations - horizontal and vertical - allow us to predict future positions of the projectile.

Muzzle velocity refers to the speed of a projectile at the moment it leaves the barrel of a gun. Calculating this is crucial in understanding and analyzing the motion of projectiles.
From our problem, we deduced the formula for vertical motion, \(y = 0.5g(x/v_{xi})^2\), which simplifies to \(y = Ax^2\). Here, \(A = 0.5g / v_{xi}^2\). To find the muzzle velocity \(v_{xi}\), we need to express it in terms of known quantities:1. Substitute \(A = y/x^2\) into the expression: - \(A = 0.5g / v_{xi}^2\)2. Rearrange the formula to solve for \(v_{xi}\): - \(v_{xi} = \sqrt{0.5g(x^2/y)}\)This rearrangement allows you to calculate the muzzle velocity when you know the values \(x\), \(y\), and \(g\). For example, if our horizontal distance \(x\) is \(3.00\, \text{m}\) and vertical drop \(y\) is \(0.21\, \text{m}\), with gravity \(g = 9.8 \text{m/s}^2\), you can directly substitute these into the equation to find \(v_{xi}\). Determining muzzle velocity this way helps in predicting how a projectile will behave.