Jack and Jill ran up the hill at 3.0 m/s. The horizontal component of Jill's velocity vector was 2.5 m/s. What was the vertical component of Jill's velocity?

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Identify the given values: The magnitude of Jill's velocity vector is 3.0 m/s, and the horizontal component of her velocity is 2.5 m/s.

Recall the Pythagorean theorem, which relates the components of a velocity vector to its magnitude: \( v^2 = v_x^2 + v_y^2 \), where \( v \) is the magnitude of the velocity, \( v_x \) is the horizontal component, and \( v_y \) is the vertical component.

Rearrange the equation to solve for the vertical component \( v_y \): \( v_y = \sqrt{v^2 - v_x^2} \).

Substitute the given values into the equation: \( v_y = \sqrt{(3.0 \ \text{m/s})^2 - (2.5 \ \text{m/s})^2} \).

Simplify the expression under the square root to find the vertical component of Jill's velocity.

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주요 개념

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Velocity Components

Velocity is a vector quantity that has both magnitude and direction. In two-dimensional motion, the velocity can be broken down into horizontal and vertical components. The horizontal component represents motion along the x-axis, while the vertical component represents motion along the y-axis. Understanding these components is essential for analyzing motion in different directions.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In the context of velocity, this theorem is used to calculate the resultant velocity when both horizontal and vertical components are known.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In physics, these functions are often used to resolve vectors into their components. For example, the sine function can be used to find the vertical component of a velocity vector when the angle and the magnitude of the vector are known, which is crucial for solving problems involving inclined motion.