Question

In a certain track and field event, the shotput has a mass of 7.30 kg and is released with a speed of 15.0 m/s at 40.0\(^\circ\) above the horizontal over a competitor's straight left leg. What are the initial horizontal and vertical components of the momentum of this shotput?

Short answer

The initial horizontal momentum is approximately 83.98 kg m/s, and the vertical momentum is approximately 70.41 kg m/s.

Step-by-step solution

Identify Given Information

The mass (m) of the shotput is 7.30 kg, the speed (v) is 15.0 m/s, and the angle above horizontal (\( \theta \)) is 40.0 degrees. We need to find the horizontal and vertical components of momentum.

Calculate Initial Speed Components

To find the components of the speed, use trigonometric functions:Horizontal speed \( v_x = v \cdot \cos(\theta) = 15.0 \cdot \cos(40^\circ) \).Vertical speed \( v_y = v \cdot \sin(\theta) = 15.0 \cdot \sin(40^\circ) \).

Calculate Horizontal Momentum Component

Momentum in the horizontal direction (\( p_x \)) is given by the product of mass and horizontal speed:\[ p_x = m \cdot v_x = 7.30 \cdot 15.0 \cdot \cos(40^\circ) \].Calculate \( p_x \) using \( \cos(40^\circ) \approx 0.766 \):\[ p_x \approx 7.30 \cdot 15.0 \cdot 0.766 = 83.98 \text{ kg m/s (approx.)} \].

Calculate Vertical Momentum Component

Momentum in the vertical direction (\( p_y \)) is given by the product of mass and vertical speed:\[ p_y = m \cdot v_y = 7.30 \cdot 15.0 \cdot \sin(40^\circ) \].Calculate \( p_y \) using \( \sin(40^\circ) \approx 0.643 \):\[ p_y \approx 7.30 \cdot 15.0 \cdot 0.643 = 70.41 \text{ kg m/s (approx.)} \].

Key concepts

Projectile motion

Projectile motion is a fascinating concept that describes the flight of an object thrown or projected into the air, which is subject to gravity and air resistance. In the context of the shotput example, the object follows a curved path as it travels both horizontally and vertically after being released. This type of motion is characterized by two independent sets of components: the horizontal motion and the vertical motion.

In this event, the shotput has an initial speed and is released at an angle, allowing it to travel a certain distance and height. The horizontal motion continues constant because of the lack of horizontal forces, while vertical motion is influenced by gravity, acting as a force that pulls the object downward.

• Horizontal motion: Constant velocity, given no air resistance.
• Vertical motion: Accelerated due to gravity, which affects height and time of flight.

Understanding projectile motion helps in determining various factors such as range, maximum height, and flight time, which are crucial for optimizing performance in events like shotput.

Trigonometry in physics

Trigonometry plays a vital role in physics, especially when analyzing problems involving angles, such as the shotput problem. It involves using trigonometric functions to break down vectors into components that are easier to manage. In the case of the shotput being released at a 40-degree angle, trigonometry allows us to separate the projectile’s initial velocity into horizontal and vertical components.

To do this, we use:

• Cosine function: To find the horizontal component of a vector or velocity (\( v_x = v \cdot \cos(\theta) \)).
• Sine function: To find the vertical component (\( v_y = v \cdot \sin(\theta) \)).

By applying these functions, it's possible to calculate exact momentum components for both directions, giving a full picture of the shotput’s movement right after it is released. This approach is essential in accurately describing motion in physics.

Vectors in physics

Vectors are fundamental in physics because they represent quantities that have both a magnitude and a direction. This means vectors can efficiently describe various physical quantities, such as velocity, force, and momentum. In the scenario of a shotput event, the initial velocity is a vector that needs to be divided into manageable components for calculation purposes.

Important characteristics of vectors include:

• Magnitude: The size or length of the vector, often representing speed or strength of a force.
• Direction: Which way the vector is pointing, crucial for understanding how an object moves.

In physics problems like the shotput, vector decomposition helps us analyze each part of the motion separately. By exploring each component (horizontal and vertical), physicists are capable of resolving the complex movement of projectiles into simpler, comprehensible parts. Understanding vectors is key for solving multi-dimensional physics problems efficiently.