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. (c) How far did she throw the shot horizontally?

Verified step by step guidance

1. Resolve the initial velocity of 12.0 m/s into horizontal and vertical components. Use the equations: \(v_{x} = v \cos(\theta)\) and \(v_{y} = v \sin(\theta)\), where \(v\) is the initial velocity and \(\theta\) is the angle above the horizontal.
2. Calculate the horizontal component of the initial velocity using the angle 51.0°: \(v_{x} = 12.0 \cos(51.0^\circ)\).
3. Since there is no acceleration in the horizontal direction (ignoring air resistance), the horizontal velocity remains constant throughout the motion.
4. The total time of flight is given as 2.08 seconds. Use this time to find the horizontal distance traveled. The horizontal distance \(x\) can be calculated using the formula: \(x = v_{x} \times t\), where \(t\) is the time of flight.
5. Substitute the calculated horizontal velocity and the time into the distance formula to find the horizontal distance the shot was thrown.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational force. It can be analyzed in two dimensions: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravity, leading to a parabolic trajectory. Understanding this concept is crucial for solving problems involving objects thrown at an angle.

Horizontal Range

The horizontal range of a projectile is the total horizontal distance it travels before hitting the ground. It can be calculated using the horizontal component of the initial velocity and the total time of flight. The formula for horizontal range is given by R = Vx * t, where Vx is the horizontal velocity and t is the time of flight. This concept is essential for determining how far the shot putter threw the shot horizontally.

Components of Velocity

When an object is launched at an angle, its initial velocity 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 found using Vy = V * sin(θ), where V is the initial velocity and θ is the launch angle. Understanding these components is vital for analyzing projectile motion and calculating distances.