91Ó°ÊÓ

To understand how far a bullet will travel horizontally before it hits the target, we need to calculate its time of flight. The bullet travels horizontally with a constant velocity given by the rifle's muzzle speed, which in this scenario is 460 m/s. The formula to find the time of flight is \[ t = \frac{d}{v} \] where \(t\) is the time, \(d\) is the distance to the target, and \(v\) is the muzzle speed of the bullet. For the given problem, with the target at 45.7 meters away, and the speed of 460 meters per second, we substitute these values into the formula: \[ t = \frac{45.7 \text{ m}}{460 \text{ m/s}} \approx 0.099 \text{ seconds} \] This means the bullet will be in the air for about 0.099 seconds before reaching the target.

While the bullet is traveling towards the target, gravity pulls it downwards. This downward movement is known as vertical displacement. We can calculate the vertical drop by using the formula for vertical displacement under constant acceleration, which in this case is due to gravity: \[ y = \frac{1}{2}gt^2 \] Here, \(g\) represents the acceleration due to gravity (approximately 9.8 m/s²), and \(t\) is the time of flight, which we previously calculated as 0.099 seconds. Substituting these values in, we get: \[ y = \frac{1}{2} \times 9.8 \text{ m/s}^2 \times (0.099 \text{ s})^2 \approx 0.048 \text{ m} \] Thus, during its flight, the bullet will fall approximately 0.048 meters (or 4.8 centimeters) due to gravity.

The concept of constant velocity is crucial in calculating the time of flight for projectiles. When an object moves at constant velocity, it covers equal distances in equal intervals of time, without any change in speed or direction. In our specific problem, the bullet exits the rifle with a constant velocity of 460 meters per second. This means there is no horizontal acceleration acting on the bullet once it exits the rifle. Because the velocity is constant, we can use the simple formula for distance traveled at constant velocity: \[ d = vt \] This relationship was used in the initial step to simplify the calculation of the bullet's time of flight. By understanding that the bullet's horizontal motion remains uniform (unaffected by forces speaking horizontally), we focus solely on vertical factors, such as gravity, to determine other elements like vertical displacement.