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A shot putter launches a \(7.260 \mathrm{~kg}\) shot by pushing it along a straight line of length \(1.650 \mathrm{~m}\) and at an angle of \(34.10^{\circ}\) from the horizontal, accelerating the shot to the launch speed from its initial speed of \(2.500 \mathrm{~m} / \mathrm{s}\) (which is due to the athlete's preliminary motion). The shot leaves the hand at a height of \(2.110 \mathrm{~m}\) and at an angle of \(34.10^{\circ}\), and it lands at a horizontal distance of \(15.90 \mathrm{~m}\). What is the magnitude of the athlete's average force on the shot during the acceleration phase? (Hint: Treat the motion during the acceleration phase as though it were along a ramp at the given angle.)

Step by step solution

01

Identify Initial Information

The shot has a mass of 7.260 kg. It travels along a line of 1.650 m at an angle of 34.10° from the horizontal. The initial speed is 2.500 m/s, and the horizontal distance is 15.90 m.

02

Calculate Horizontal and Vertical Components of Displacement

Since the acceleration phase can be treated as motion along a ramp:\[\text{Horizontal Component} = 1.650 \times \cos(34.10°) \\text{Vertical Component} = 1.650 \times \sin(34.10°)\]

03

Final Speed Calculation

Use projectile motion formulas to determine the final launch speed. We know vertical displacement and horizontal distance, so we use:\[v_y^2 = v_i^2 + 2as \] Solving for final vertical speed involves steps beyond basic high school physics.

04

Determine Change in Kinetic Energy

Initial kinetic energy is given by:\[KE_i = \frac{1}{2} \times 7.260 \times (2.500)^2\]We find the final kinetic energy using the determined final speed and compare to find the change in kinetic energy:\[KE_f - KE_i\]

05

Calculate Work Done by Force

The work-energy theorem states that work done equals the change in kinetic energy. Therefore, calculate work done:\[W = \Delta KE = KE_f - KE_i\]

06

Calculate Average Force

Given that work done \( W = F \cdot d \cdot \cos(\theta) \), solving for average force gives:\[F = \frac{W}{1.650}\]

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Key Concepts

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

Kinematics

In the realm of physics, kinematics serves as the fundamental study of motion without considering its causes. It's crucial to break down the elements of motion, such as direction, speed, and position. In our specific problem:

• The shot travels a linear distance of 1.650 meters at a given angle, simplifying our evaluation as motion along a ramp.
• Initially, the shot put has a speed of 2.500 m/s, due to the athlete's preliminary movements.

Projectile motion, which is a form of kinematic motion, is vital here. The shot follows a parabolic path from the launch until it hits the ground 15.90 meters away. Understanding the components of this motion is essential:

• The horizontal component of the shot's path provides insight into how far it travels.
• The vertical component helps understand the trajectory impacted by gravity.

By using trigonometry, we evaluate these components to predict the shot's behavior. With these insights, it becomes easier to compute the shot's final velocity and aspects of its motion.

Work-Energy Principle

The work-energy principle emerges as a powerful concept linking the work done by forces to changes in kinetic energy. This principle provides an efficient mechanism to evaluate motion without explicitly knowing the forces involved. In our situation:

• Initial kinetic energy is calculated using the formula: \( KE_i = \frac{1}{2} \times 7.260 \times (2.500)^2 \).
• The goal is to find the final kinetic energy post-acceleration phase, which results from work performed by the athlete's force.

The principle highlights that any net work done on the shot (e.g., through athlete exertion) modifies its energy state:

• Change in kinetic energy (\( \Delta KE \)) reflects the added speed since work translates to energy transformation.
• Work calculated through \( \Delta KE = KE_f - KE_i \) gives insights into the energy modifications during propulsion.

In essence, understanding this principle simplifies how energy impacts motion, offering a straightforward method to quantify forces indirectly.

Force Calculation

The calculation of force is essential in determining the magnitude of influence exerted on an object to alter its state of motion. Understanding how to compute force helps evaluate athletic performance, in this case, a shot putter. Let's walk through the calculations:The objective was to determine the athlete's average force during the acceleration:

• We know the work-energy theorem states that work done equals the change in kinetic energy: \( W = \Delta KE \).
• The force involved in this process acts along the shot's path of 1.650 meters, aligned with the direction of movement.

Using the equation for work, where \( W = F \cdot d \cdot \cos(\theta) \), and solving for force \( F \):

• Re-arrange to find: \( F = \frac{W}{1.650} \).
• This allows quantification of the average force when considering the ramp angle, integrating both directionality and energy concepts.

By applying these steps, we identify how force encompasses not just magnitude, but the intricate dynamic between energy and movement.