91Ó°ÊÓ

When we talk about muzzle speed, we're referring to the initial speed at which a bullet or projectile exits the barrel of a gun. This speed is crucial in determining how far and how fast the projectile will travel. In our exercise, the muzzle speed is given as \(670 \, \mathrm{m/s}\), which is quite high and indicates the bullet moves very swiftly. The speed with which a bullet leaves the gun affects both the vertical motion (due to gravity) and the horizontal motion (cross distance). It's the starting force behind the projectile’s journey. Some important points to remember about muzzle speed:

• Higher muzzle speed usually results in a longer range for the projectile.
• It is one of the initial conditions used to solve projectile motion problems.
• This speed is determined by the gun's design, the type of bullet, and the amount of gunpowder used.

Understanding muzzle speed is essential to make predictions about the bullet's behavior after it leaves the gun.

Vertical motion in projectile physics refers to how gravity influences the path of the projectile (the bullet, in our case) after it is fired. Even when aimed horizontally, a bullet will drop as it travels due to gravitational pull. The vertical motion is independent of the horizontal motion and can be described with the equation \( y = \frac{1}{2} g t^2 \). Here, \( y \) is the vertical distance, \( g \) is the acceleration due to gravity (\(9.81 \, \mathrm{m/s^2}\)), and \( t \) is the time in seconds.In our exercise, the bullet drops \(0.025 \, \mathrm{m}\) before hitting the target. By substituting known values into the vertical motion equation, we can find the time it takes for the bullet to hit the target:\[ 0.025 = \frac{1}{2} \times 9.81 \times t^2 \]This demonstrates how vertical acceleration due to gravity is a key factor affecting where the bullet will make contact with the target. Gravity acts as a constant force, pulling the bullet down over its flight. Knowing this helps us understand why aiming slightly above targets can compensate for this natural drop.

Once we know how long it takes for the projectile (bullet) to travel, we can find out how far it travels horizontally during that time. This is called the horizontal distance, and it's what we calculated in the bullet's journey to the bull's-eye target. The formula used here is \( x = v_0 t \), where \( x \) is the horizontal distance, \( v_0 \) is the muzzle speed, and \( t \) is the time calculated from the vertical motion.For our exercise:

• The muzzle speed \( v_0 \) is \(670 \, \mathrm{m/s}\).
• The time of flight \( t \) was calculated as approximately \(0.0715 \, \mathrm{s}\) from solving the vertical motion equation.
• Using these values, we calculate the horizontal distance: \( x = 670 \, \mathrm{m/s} \times 0.0715 \, \mathrm{s} \), which gives us \( x \approx 47.90 \, \mathrm{m}\).

This calculation shows how effectively the bullet covers a substantial distance in a short time. Horizontal distance in projectile motion is impacted by the duration of flight and the initial muzzle speed, demonstrating how these elements of physics intertwine to determine the projectile's travel path.