Problem 27 - Thomas Calculus

Question

Projectile motion A gun with muzzle velocity of 1200 \(\mathrm{ft} / \mathrm{sec}\) is fired at an angle of \(8^{\circ}\) above the horizontal. Find the horizontal and vertical components of the velocity.

Step-by-Step Solution

Answer

Horizontal component: \( 1188.36 \mathrm{ft/sec} \); Vertical component: \( 167.04 \mathrm{ft/sec} \).

Step 1: Determine Horizontal Component

The horizontal component of the velocity can be calculated using the formula for horizontal velocity: \( v_{x} = v \, \cos(\theta) \), where \( v \) is the muzzle velocity and \( \theta \) is the angle of projection. Substitute \( v = 1200 \, \mathrm{ft/sec} \) and \( \theta = 8^{\circ} \) into the formula: \( v_{x} = 1200 \, \cos(8^{\circ}) \).

Step 2: Calculate Horizontal Component

Calculate \( \cos(8^{\circ}) \) using a calculator to find \( \cos(8^{\circ}) \approx 0.9903 \). Thus, the horizontal component is \( v_{x} = 1200 \, \times \, 0.9903 \approx 1188.36 \mathrm{ft/sec} \).

Step 3: Determine Vertical Component

The vertical component of the velocity is calculated using the formula for vertical velocity: \( v_{y} = v \, \sin(\theta) \). Substitute \( v = 1200 \, \mathrm{ft/sec} \) and \( \theta = 8^{\circ} \) into the formula: \( v_{y} = 1200 \, \sin(8^{\circ}) \).

Step 4: Calculate Vertical Component

Calculate \( \sin(8^{\circ}) \) using a calculator to find \( \sin(8^{\circ}) \approx 0.1392 \). Thus, the vertical component is \( v_{y} = 1200 \, \times \, 0.1392 \approx 167.04 \mathrm{ft/sec} \).

Key Concepts

Horizontal Component of Velocity

When an object is projected at an angle, the horizontal component of its velocity is one of the critical factors in understanding its motion. Essentially, this component tells us how quickly the object moves along the horizontal axis. To find it, we use the formula: \[ v_{x} = v \cos(\theta) \]where \( v \) is the initial velocity and \( \theta \) is the angle of projection. This equation highlights the role of the trigonometric function cosine.
By plugging in the given values—like a muzzle velocity of 1200 ft/sec at an angle of \( 8^{\circ} \)—we can use a calculator to find \( \cos(8^{\circ}) \approx 0.9903 \). Thus, the horizontal velocity is computed as 1188.36 ft/sec. This means the projectile will travel this distance horizontally each second, unaffected by gravity. Remember: the horizontal component remains constant when there's no air resistance, as there are no forces acting horizontally.

Vertical Component of Velocity

While the horizontal component determines the distance traveled along the x-axis, the vertical component of velocity affects how high and how long the projectile will stay in the air. The formula used here is: \[ v_{y} = v \sin(\theta) \]Again, \( v \) is the initial velocity and \( \theta \) is the angle of projection. The sine function from trigonometry helps in calculating this part effectively.
If we substitute \( v = 1200 \, \mathrm{ft/sec} \) and \( \theta = 8^{\circ} \), and find that \( \sin(8^{\circ}) \approx 0.1392 \), the vertical component calculates to about 167.04 ft/sec. Unlike the horizontal component, the vertical velocity changes over time due to gravity pulling the projectile downward. This will slow the projectile's upward motion until it reaches its highest point and then speeds it up as it falls back down.

Trigonometric Functions

Trigonometric functions are the mathematical foundation for breaking down velocity into its horizontal and vertical components. They use angles to relate the sides of a right triangle, which is crucial for projectile motion. There are three primary functions: sine, cosine, and tangent.
For projectile motion:

• Cosine is used to calculate the horizontal component. It measures the length of the adjacent side over the hypotenuse in a right triangle.
• Sine is used to find the vertical component. It measures the length of the opposite side over the hypotenuse.

In our example, these functions convert the angle of 8° into values that help express how fast the projectile moves horizontally and vertically. When dealing with such calculations, it's often handy to use calculators for obtaining accurate sine and cosine values for given angles. Understanding these functions opens the door to solving many physics problems involving angled projections.