91Ó°ÊÓ

In shot putting, many athletes elect to launch the shot at an angle that is smaller than the theoretical one (about \(42^{\circ}\) ) at which the distance of a projected ball at the same speed and height is greatest. One reason has to do with the speed the athlete can give the shot during the acceleration phase of the throw. Assume that a \(7.260 \mathrm{~kg}\) shot is accelerated along a straight path of length \(1.650 \mathrm{~m}\) by a constant applied force of magnitude \(380.0 \mathrm{~N}\), starting with an initial speed of \(2.500 \mathrm{~m} / \mathrm{s}\) (due to the athlete's preliminary motion). What is the shot's speed at the end of the acceleration phase if the angle between the path and the horizontal is (a) \(30.00^{\circ}\) and (b) \(42.00^{\circ}\) ? (Hint: Treat the motion as though it were along a ramp at the given angle. \()\) (c) By what percent is the launch speed decreased if the athlete increases the angle from \(30.00^{\circ}\) to \(42.00^{\circ} ?\)

Step by step solution

01

Determine the Work Done

The work done by the force along the ramp is calculated by the formula:\[ W = F \cdot d \cdot \cos(\theta) \] where \( F = 380.0 \, \text{N} \), \( d = 1.650 \, \text{m} \), and \( \theta \) is the angle of the incline.Let's find the work done for both angles.For \( \theta = 30.00^{\circ} \):\[ W_{30} = 380.0 \times 1.650 \times \cos(30.00^{\circ}) \approx 543.467 \, \text{J} \]For \( \theta = 42.00^{\circ} \):\[ W_{42} = 380.0 \times 1.650 \times \cos(42.00^{\circ}) \approx 465.312 \, \text{J} \]

02

Apply Work-Energy Principle

The work-energy principle states that the work done is equal to the change in kinetic energy:\[ W = \Delta KE = \frac{1}{2}m(v^2 - u^2) \]where \( m = 7.260 \, \text{kg} \), \( u = 2.500 \, \text{m/s} \), and \( v \) is the final speed.For \( \theta = 30.00^{\circ} \):\[ 543.467 = \frac{1}{2} \times 7.260 \times (v_{30}^2 - 2.500^2) \]Solving for \( v_{30} \):\[ v_{30}^2 = \frac{2 \times 543.467}{7.260} + 2.500^2 \approx 85.411 \v_{30} \approx 9.243 \, \text{m/s} \]

03

Calculate Final Speed for Second Angle

Now, apply the same work-energy equation for \( \theta = 42.00^{\circ} \):\[ 465.312 = \frac{1}{2} \times 7.260 \times (v_{42}^2 - 2.500^2) \]Solving for \( v_{42} \):\[ v_{42}^2 = \frac{2 \times 465.312}{7.260} + 2.500^2 \approx 75.940 \v_{42} \approx 8.712 \, \text{m/s} \]

04

Calculate Percent Decrease in Launch Speed

The percentage decrease in speed from \( v_{30} \) to \( v_{42} \) is given by:\[ \%\text{ decrease} = \left(\frac{v_{30} - v_{42}}{v_{30}}\right) \times 100\% \]Substituting the values, we get:\[ \%\text{ decrease} = \left(\frac{9.243 - 8.712}{9.243}\right) \times 100\% \approx 5.74\% \]

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Key Concepts

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

Work-Energy Principle

The Work-Energy Principle is a powerful concept in physics that connects the work done on an object to its change in kinetic energy. Simply put, when work is done on an object, it changes the object’s energy state, usually increasing or decreasing its speed.
Let's break down the major components:

• **Work Done (W):** This is calculated using the formula: \(W = F \cdot d \cdot \cos(\theta)\), where \(F\) is the force, \(d\) is the distance, and \(\theta\) is the angle between the force and the direction of movement.
• **Change in Kinetic Energy (\(\Delta KE\)):** This reflects how the velocity of the object changes due to the applied force. It's given by \(\Delta KE = \frac{1}{2}m(v^2 - u^2)\), where \(v\) is the final speed, \(u\) is the initial speed, and \(m\) is the mass.

These principles are essential in understanding how energy transformation leads to motion changes. They provide us with the tools to calculate how different forces and distances affect the motion of an object through changes in speed and direction.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It's all about the geometric aspects of motion.
There are a few key ideas in kinematics:

• **Displacement:** A vector quantity that describes the change in position of an object.
• **Velocity:** A vector quantity that tells us the speed and direction of the object's motion. We compute velocity as \(v = u + at\), where \(a\) is acceleration and \(t\) is time.
• **Acceleration:** The rate at which an object's velocity changes over time, expressed as \(a = \frac{\Delta v}{\Delta t}\).

In the context of projectile motion, kinematics helps us determine important aspects like the maximum height an object reaches and the time it remains in the air. It’s a critical concept in predicting the trajectory and final landing point of projectiles.

Inclined Planes

Inclined planes are flat surfaces tilted at an angle relative to the horizontal. They form a simple machine that makes it easier to raise or lower objects, which is a concept frequently seen in physics.
Here are some of the basics of inclined planes:

• **Angle of Incline (\(\theta\)):** This is the angle between the inclined surface and the horizontal base. It affects the components of force acting on an object moving along the plane.
• **Forces on Incline:** Forces include the gravitational force, which can be decomposed into components parallel and perpendicular to the plane. The normal force acts perpendicular to the surface, and friction might act parallel but in the opposite direction of movement.

On inclined planes, the motion can be analyzed using Newton's laws, and more specifically, using kinematic equations adapted to account for the incline angle. This setup plays a key role in understanding how angles affect motion and speed, particularly in sports activities like shot putting.

Angle of Projection

The Angle of Projection is a critical factor in determining the trajectory and range of a projectile. It is the angle at which an object is launched, relative to the horizontal.
Key insights into the angle of projection include:

• **Optimal Angle:** For maximum range in a vacuum, this angle is typically \(45^{\circ}\). However, in real-world scenarios, factors like air resistance and initial heights may alter the ideal angle.
• **Effect on Range and Height:** By changing the angle, an athlete can either maximize the height reached or the horizontal distance covered. This is essential for sports like javelin throw or shot put, where athletes must decide whether to prioritize distance over height.

The angle of projection is fundamental in physics problems dealing with projectile motion since it directly influences the equations for horizontal and vertical motion. Understanding how to manipulate this angle allows athletes and engineers to optimize outcomes based on their specific goals and situational needs.