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

검증된 단계별 안내

Step 1: Understand the problem. The velocity vector of the cannonball is given as v = (75î + 45ĵ), where î represents the horizontal component and ĵ represents the vertical component. The goal is to find the angle θ that the barrel is tilted above the horizontal.

Step 2: Recall the formula for the angle of a vector relative to the horizontal axis. The angle θ can be calculated using the tangent function: tan(θ) = (vertical component) / (horizontal component). In this case, tan(θ) = 45 / 75.

Step 3: Use the inverse tangent function (arctan or tan⁻¹) to find the angle θ. The formula is θ = tan⁻¹(45 / 75). This will give the angle in radians or degrees, depending on the calculator or method used.

Step 4: Ensure the units are consistent. Since the components of the velocity vector are both in the same units, no conversion is necessary before applying the formula.

Step 5: Interpret the result. The angle θ represents the tilt of the barrel above the horizontal. If needed, convert the angle from radians to degrees using the formula: degrees = radians × (180/π).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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, specifically through the tangent function: θ = arctan(v_y/v_x), where v_y and v_x are the vertical and horizontal components of the velocity. 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 determine relationships between the angle of projection and the components of velocity. Mastery of these functions is necessary for solving problems involving angles and distances in physics.