Problem 94 - Precalculus with Limits

Question

A sharpshooter intends to hit a target at a distance of \( 1000 \) yards with a gun that hasa muzzle velocity of \( 1200 \) feet per second (see figure).Neglecting air resistance, determine the gun minimum angle of elevation \( \theta \) if the range \( r \) is given by \( r = \dfrac{1}{32} v_0^2 \sin 2 \theta \)

Step-by-Step Solution

Answer

The sharpshooter has to aim at a minimum angle of \(\theta\) (insert value obtained in Step 5) in order to hit the target.

Step 1: Convert units of range

The distance to the target is given in yards, while the velocity is given in feet per second. These units need to be the same. Since 1 yard is equal to 3 feet, convert 1000 yards to 3000 feet.

Step 2: Substitute the values into the equation

Substitute \(r = 3000\) feet and \(v_0 = 1200\) feet per second into the equation \(r = \frac{1}{32} v_0^2 \sin 2 \theta\). This gives us \(3000 = \frac{1}{32} * (1200)^2 * \sin(2\theta)\).

Step 3: Solve for \(\sin(2\theta)\)

Next, solve the equation for \(\sin(2\theta)\). Divide both sides of the equation by \(\frac{1}{32} * (1200)^2\) to obtain \(\sin(2\theta) = \frac{3000}{\frac{1}{32} * (1200)^2}\).

Step 4: Find the value of \(2\theta\)

Find \(2\theta\) by using the arcsine function, also written as \(\sin^{-1}\), on the value obtained for \(\sin(2\theta)\). This will give you \(2\theta\).

Step 5: Find the value of \(\theta\)

Finally, divide the value of \(2\theta\) by 2 to find the angle of elevation \(\theta\). This is the minimum angle necessary to hit the target.

Key Concepts

Projectile Motion

When an object is thrown into the air at an angle, it follows a curved path before landing. This type of movement is known as projectile motion, which is a fundamental concept in physics, particularly in the study of mechanics. Projectile motion can be analyzed using two independent components - horizontal and vertical motion.
Projectile motion is subject to the forces of gravity, which is why objects eventually fall to the ground. By analyzing the motion using these two components, one can predict the trajectory of the object and determine important parameters such as range, maximum height, time of flight, and the angle of launch required to hit a specific target.

Key Factors Influencing Projectile Motion

• Initial Velocity (v₀): The speed at which the projectile is launched.
• Angle of Elevation (θ): The angle at which the projectile is launched above the horizontal.
• Gravity: The constant force that pulls the projectile downwards.
• Air Resistance: Usually assumed to be negligible in simple calculations, but can significantly alter the projectile's path in real-world scenarios.

The equation provided in the exercise is a simplified version, assuming no air resistance and a level ground. It calculates the range based on the initial velocity and the launch angle.

Trigonometry in Physics

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. Its principles are widely applied in physics, especially when dealing with angles, waves, and oscillations.
In the context of projectile motion, trigonometry is used to decompose the projectile's velocity into horizontal and vertical components. This is crucial for understanding the motion's direction and for calculating various attributes of the projectile's trajectory, such as range and maximum height.
The sine function, in particular, which relates the ratio of the opposite side to the hypotenuse of a right-angled triangle, is used in the equation for calculating the range of the projectile. Specifically, the equation incorporates the sine of double the launch angle, denoted as sin(2θ), which is a key example of how trigonometry is essential in analyzing cases of projectile motion in physics.

Solving Trigonometric Equations

Trigonometric equations involve trigonometric functions such as sine, cosine, and tangent. Solving these equations is a process of finding the angles that satisfy the equation given a particular ratio of sides in a right-angled triangle or, in the context of physics, specific motion attributes.
The solution provided for the sharpshooter's problem involves finding the angle of elevation that makes the projectile hit a target at a certain distance. To solve for the angle, one needs to manipulate the equation algebraically and then use inverse trigonometric functions — in this case, the arcsine (sin⁻¹) function.
Inverse trigonometric functions are used to find angle measures when the value of a trigonometric function is known. For the sharpshooter's problem, after isolating sin(2θ) algebraically, we use the arcsine function to find the measure of 2θ, which is then halved to find the angle of elevation θ. In real-world scenarios, calculators or computer programs are used to perform these calculations as they can provide accurate results swiftly.