91Ó°ÊÓ

A high-powered rifle fires a bullet with a muzzle speed of \(1.00 \mathrm{km} / \mathrm{s} .\) The gun is pointed horizontally at a large bull's eye target-a set of concentric rings- \(200 \mathrm{m}\) away. (a) How far below the extended axis of the rifle barrel does a bullet hit the target? The rifle is equipped with a telescopic sight. It is "sighted in" by adjusting the axis of the telescope so that it points precisely at the location where the bullet hits the target at \(200 \mathrm{m}\). (b) Find the angle between the telescope axis and the rifle barrel axis. When shooting at a target at a distance other than \(200 \mathrm{m}\) the marksman uses the telescopic sight, placing its crosshairs to "aim high" or "aim low" to compensate for the different range. Should she aim high or low, and approximately how far from the bull's eye, when the target is at a distance of (c) \(50.0 \mathrm{m},\) (d) \(150 \mathrm{m},\) or (e) \(250 \mathrm{m} ?\) Note: The trajectory of the bullet is everywhere so nearly horizontal that it is a good approximation to model the bullet as fired horizontally in each case. What if the target is uphill or downhill? (f) Suppose the target is \(200 \mathrm{m}\) away, but the sight line to the target is above the horizontal by \(30^{\circ} .\) Should the marksman aim high, low, or right on? (g) Suppose the target is downhill by \(30^{\circ} .\) Should the marksman aim high, low, or right on? Explain your answers.

Step by step solution

01

Calculate The Vertical Displacement

Using the standard equation for motion under gravity, the vertical displacement \(y\) covered by the bullet in time \(t\) is given by \(y = \frac{1}{2}gt^2\). The horizontal range \(x = vt\), thus the time \(t\) to target can be computed as \(t = \frac{x}{v}\), where \(v = 1km/s = 1000m/s\) and \(x = 200m\). Thus, substituting time into the equation for vertical displacement, we get \(y = \frac{1}{2}g(\frac{x}{v})^2\).

02

Calculate The Scope Angle

The angle \(\theta\) between the telescope axis and the rifle barrel axis can be computed using tan function of the ratio between vertical displacement \(y\) and horizontal distance \(x\), i.e. \(\theta = tan^{-1}(\frac{y}{x})\).

03

Aim Adjustment for Different Ranges

On analyzing the bullet’s trajectory, it is clear that for targets at less than 200m, the aim needs to be higher than the bull's eye and for targets more than 200m, the aim needs to be lowered. By using the displacement calculation as in Step 1, the deviation from the target can be computed to indicate how high or low to aim at 50m, 150m and 250m.

04

Uphill or Downhill Target

With reference to the tan function, if we add or remove angle component equivalent to uphill or downhill slope, we can accordingly decide whether to aim high or low. For uphill aim generally needs to be slightly higher according to the slope whereas for downhill aim has to be slightly lower.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Projectile Motion

Projectile motion describes the motion of an object that is thrown or projected into the air, subject to only the acceleration of gravity. The path that the object follows is known as its trajectory. In our scenario, a bullet fired from a high-powered rifle is considered a projectile. Since the rifle is aimed horizontally, the only force acting on the bullet once it leaves the barrel is gravity (ignoring air resistance).

When thinking about projectile motion, there are two key components to consider: horizontal and vertical motions. These two motions are independent of each other. The horizontal component is governed by the initial velocity of the bullet, which in the case of our exercise is a substantial 1 km/s. The bullet will maintain this horizontal speed throughout its flight because there are no horizontal forces acting on it after it's been fired. On the other hand, the vertical motion is governed by the gravity of Earth, pulling the projectile downward at an acceleration of approximately 9.81 m/s^2.

Motion Under Gravity

The term 'motion under gravity' refers to the movement of any object under the sole influence of gravity. This is a fundamental concept in physics, as it explains how objects behave when dropped, thrown, or launched horizontally, such as the bullet from our exercise. Under the influence of gravity, all objects will accelerate downward at the same rate, regardless of their mass.

This acceleration due to gravity, denoted by 'g', is constant near the Earth's surface and has a value of approximately 9.81 m/s^2. The downward path of the projectile is parabolic because the horizontal velocity is constant whereas the vertical velocity increases linearly with time due to the constant acceleration imparted by gravity.

Telescopic Sight Adjustment

In our exercise, the telescopic sight of a rifle is an essential tool for precision shooting. The primary function of the telescopic sight is to align the shooter's aim with the point of impact of the bullet. By 'sighting in' the scope, shooters can ensure their aim is accurate at a specific distance.

However, when the target distance changes, the shooter must adjust their aim because the bullet drops over longer distances due to gravity. This is where understanding the ballistic trajectory is crucial. At distances less than the sighted range, the bullet will impact above the line of sight, and so the shooter must aim lower ('aim high'). Conversely, at distances greater than the sighted range, the shooter must raise the aim point ('aim low') to compensate for the increased bullet drop.

Range of a Projectile

The range of a projectile is defined as the horizontal distance it travels before hitting the ground. For a bullet fired from a gun, the range is critical for marksmen to calculate where the bullet will fall in relation to the target. In a controlled environment, when firing a bullet horizontally at a known velocity, as in our exercise, the range can be predicted using the equations of projectile motion.

For a projectile fired at an angle, the range would depend on both the speed and angle of release. The range is longest when a projectile is fired at a 45-degree angle assuming a flat surface with no air resistance. However, in reality, air resistance, elevation changes, and other factors can significantly affect the range. Marksmen can use information about the range to make adjustments for targets at different distances by considering how much lower or higher they need to aim.