A cannonball leaves the barrel with velocity v = (75î ＋ 45ĵ). At what angle is the barrel tilted above horizontal?

Verified step by step guidance

1. Identify the components of the velocity vector. Here, the horizontal component (v_x) is 75 m/s and the vertical component (v_y) is 45 m/s.
2. Understand that the angle \( \theta \) of the barrel above the horizontal can be found using the tangent function, which relates the opposite side to the adjacent side in a right triangle. In this context, the opposite side is the vertical component and the adjacent side is the horizontal component of the velocity.
3. Set up the equation using the tangent function: \( \tan(\theta) = \frac{v_y}{v_x} \).
4. Substitute the known values into the equation: \( \tan(\theta) = \frac{45}{75} \).
5. Solve for \( \theta \) by taking the arctangent (inverse tangent) of both sides: \( \theta = \tan^{-1}(\frac{45}{75}) \). This will give you the angle in degrees.

Key Concepts

Velocity Vector

Velocity is a vector quantity that describes the rate of change of an object's position. It has both magnitude and direction, represented in this case by the components v = (75î + 45ĵ), where î and ĵ are unit vectors in the horizontal and vertical directions, respectively. Understanding how to interpret and manipulate velocity vectors is crucial for analyzing projectile motion.

Angle of Projection

The angle of projection is the angle at which an object is launched relative to the horizontal axis. It can be calculated using the components of the velocity vector. The angle θ can be found using the tangent function: θ = arctan(v_y/v_x), where v_y and v_x are the vertical and horizontal components of the velocity, respectively. This angle is essential for predicting the trajectory of the projectile.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. In the context of projectile motion, these functions are used to resolve velocity vectors into their components and to calculate angles. Mastery of these functions is necessary for solving problems involving angles and distances in physics.