To determine how high above the deer the sportsman should aim his rifle, we need to analyze the situation using some physics principles, particularly those related to projectile motion. Let's break this down step by step.
Understanding the Problem
The deer is located at a distance of \(10 \times \sqrt{181}\) meters along the line of sight, and the height of the hill is 90 meters. The gun has a muzzle velocity of 100 m/s. The goal is to find out how much higher than the deer the sportsman should aim to ensure a successful shot.
Calculating the Horizontal Distance
First, we need to calculate the horizontal distance to the deer:
- Distance = \(10 \times \sqrt{181}\)
- Calculating \(\sqrt{181}\) gives approximately 13.45.
- Thus, the horizontal distance is \(10 \times 13.45 \approx 134.5\) meters.
Projectile Motion Basics
When a projectile is fired, it follows a parabolic trajectory. The key factors affecting this trajectory are:
- The initial velocity of the projectile (in this case, the bullet).
- The angle of projection.
- The acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\)).
- The height difference between the gun and the target.
Finding the Angle of Projection
To hit the deer, we need to consider the vertical drop of the bullet due to gravity while it travels horizontally. The time \(t\) it takes for the bullet to reach the deer can be calculated using the horizontal distance and the muzzle velocity:
Time \(t = \frac{\text{horizontal distance}}{\text{muzzle velocity}} = \frac{134.5}{100} \approx 1.345\) seconds.
Calculating the Vertical Drop
During this time, the bullet will drop due to gravity. The vertical drop \(d\) can be calculated using the formula:
\(d = \frac{1}{2} g t^2\)
Substituting \(g = 9.81 \, \text{m/s}^2\) and \(t \approx 1.345\) seconds:
\(d = \frac{1}{2} \times 9.81 \times (1.345)^2 \approx 8.83\) meters.
Determining the Aim Height
Now, we know that the bullet will drop approximately 8.83 meters by the time it reaches the deer. Since the hill is 90 meters high, the sportsman needs to aim above the deer to compensate for this drop:
Height above the deer = Height of the hill - Vertical drop = \(90 - 8.83 \approx 81.17\) meters.
Final Aim Adjustment
To ensure a successful shot, the sportsman should aim approximately 81.17 meters above the deer. This adjustment accounts for the bullet's drop due to gravity over the distance it travels. By aiming at this height, he increases his chances of hitting the deer accurately.
