91Ó°ÊÓ

A shot putter releases the shot some distance above the level ground with a velocity of \(12.0 \mathrm{m} / \mathrm{s}, 51.0^{\circ}\) above the horizontal. The shot hits the ground 2.08 \(\mathrm{s}\) later. You can ignore air resistance. (a) What are the components of the shot's acceleration while in flight? (b) What are the components of the shot's velocity at the beginning and at the end of its trajectory? (c) How far did she throw the shot horizontally? (d) Why does the expression for \(R\) in Example 3.8 not give the correct answer for part (c)? (e) How high was the shot above the ground when she released it? Draw \(x-t\) , \(y-t, v_{x}-t,\) and \(v_{y}-t\) graphs for the motion.

Step by step solution

01

Determine the acceleration components

The only acceleration acting on the shot putter while in flight is due to gravity. Therefore, the horizontal component of the acceleration is zero, and the vertical component is the gravitational acceleration, acting downwards.- Horizontal acceleration, \(a_x = 0\)- Vertical acceleration, \(a_y = -9.8 \, \text{m/s}^2\)

02

Calculate the initial velocity components

The initial velocity can be broken down into horizontal and vertical components using trigonometric functions.- Horizontal component of initial velocity, \(v_{0x} = v_0 \cdot \cos(51.0^{\circ}) = 12.0 \, \text{m/s} \cdot \cos(51.0^{\circ})\)- Vertical component of initial velocity, \(v_{0y} = v_0 \cdot \sin(51.0^{\circ}) = 12.0 \, \text{m/s} \cdot \sin(51.0^{\circ})\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics

Kinematics is a branch of mechanics that describes the motion of objects without considering the forces that cause this motion. In the projectile motion problem involving the shot putter, kinematics helps us analyze how the shot traverses through the air. By examining its velocity, time of flight, and acceleration, we can predict various aspects of its journey.

Understanding kinematics is crucial in projectile motion. This involves knowing the initial velocity of the shot and how it decomposes into horizontal and vertical components. Because kinematics deals with motion, it provides us with the tools to calculate:

• Velocity at different points in the trajectory.
• The time the shot stays in the air.
• The distance it travels before hitting the ground.

Vector Components

In physics, decomposing forces, velocities, and other vector quantities into their components is essential for analysis. Breaking down the shot's initial velocity into horizontal and vertical components allows us to better understand and calculate its motion.

To find these vector components, especially for projectile motion like the shot put:

• Use the angle of projection, which is provided as 51 degrees above the horizontal.
• Apply trigonometric functions: cosine for the horizontal component and sine for the vertical component.

For the shot put:

• Horizontal Velocity: Given as \( v_{0x} = 12.0 \, \text{m/s} \cdot \cos(51.0^\circ) \)
• Vertical Velocity: Calculated as \( v_{0y} = 12.0 \, \text{m/s} \cdot \sin(51.0^\circ) \)

Gravity

Gravity is a constant force that every object on Earth experiences towards the center of the planet. It provides the consistent downward acceleration seen in projectile motion problems. For the shot putter, gravity is the only force acting on the shot while it's in flight.
Here's what you need to know about its role:

• Vertical acceleration due to gravity is \( a_y = -9.8 \, \text{m/s}^2 \), indicating a downward force.
• The horizontal motion remains unaffected by gravity, hence horizontal acceleration \( a_x \) is zero.

Understanding gravity's effects helps explain why the vertical component of the object's velocity will decrease, cease at the peak, and increase again as it descends, while the horizontal component of velocity remains constant.

Physics Problems

Solving physics problems often involves applying theoretical principles to practical scenarios, like the one with the shot put. The main goal is to take known quantities and use them to find unknowns by employing equations and consistent reasoning.
In tackling this exercise, consider the following:

• Always identify what quantities are known and unknown.
• Break the problem into smaller parts. First, deal with components separately before combining them into a cohesive solution.
• Use kinematic equations carefully. Combine horizontal and vertical motion studies for accurate results.

Physics problems like these can be tricky, but understanding the core concepts of motion, force, and vector components will guide you to the correct solution. Always visualize the problem and draw diagrams if necessary to aid in understanding.