A shot putter releases the shot some distance above the level ground with a velocity of 12.0 m/s, 51.0° above the horizontal. The shot hits the ground 2.08 s later. Ignore air resistance. How far did she throw the shot horizontally?

검증된 단계별 안내

Identify the initial velocity components. The initial velocity is given as 12.0 m/s at an angle of 51.0° above the horizontal. Use trigonometry to find the horizontal and vertical components: \( v_{x} = v \cdot \cos(\theta) \) and \( v_{y} = v \cdot \sin(\theta) \), where \( v = 12.0 \) m/s and \( \theta = 51.0° \).

Calculate the horizontal component of the velocity. Substitute the values into the equation \( v_{x} = 12.0 \cdot \cos(51.0°) \) to find the horizontal velocity component.

Understand that the horizontal motion is uniform because air resistance is ignored. The horizontal distance traveled, or range, can be calculated using the formula \( d = v_{x} \cdot t \), where \( t = 2.08 \) s is the time of flight.

Substitute the horizontal velocity component and the time into the range formula: \( d = v_{x} \cdot 2.08 \). This will give you the horizontal distance the shot traveled.

Review the steps to ensure all calculations are based on the correct components and formulas. Make sure the trigonometric calculations are accurate and the time of flight is correctly applied to find the horizontal distance.

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주요 개념

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Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. It involves two components: horizontal and vertical motion. The horizontal motion occurs at a constant velocity, while the vertical motion is influenced by gravity, causing the object to follow a parabolic trajectory.

Kinematic Equations

Kinematic equations describe the motion of objects in terms of displacement, velocity, acceleration, and time. For projectile motion, these equations help determine the horizontal and vertical components of motion. The horizontal distance can be calculated using the formula: distance = velocity × time, where the horizontal velocity is the initial velocity multiplied by the cosine of the launch angle.

Components of Velocity

The initial velocity of a projectile can be broken down into horizontal and vertical components using trigonometric functions. The horizontal component (Vx) is found using Vx = V * cos(θ), and the vertical component (Vy) is Vy = V * sin(θ), where V is the initial velocity and θ is the angle of projection. These components are crucial for analyzing the projectile's motion separately in horizontal and vertical directions.