Chap6hw

1.

1. Force:
   A force is a push or pull acting on an object, which can cause it to accelerate, decelerate, or change direction. It is a vector quantity, meaning it has both magnitude and direction. The SI unit of force is the Newton (N).

2. Mass:
   Mass is a measure of the amount of matter in an object. It is a scalar quantity and is typically measured in kilograms (kg). Mass determines an object’s resistance to acceleration (inertia) and its gravitational attraction to other objects.

3. Acceleration:
   Acceleration is the rate at which an object’s velocity changes over time. It is a vector quantity and is measured in meters per second squared (m/s²). Acceleration can result from a change in speed, direction, or both.

4. Inertia:
   Inertia is the tendency of an object to resist changes in its state of motion (whether at rest or moving at a constant velocity). It is directly related to the object’s mass—the greater the mass, the greater the inertia.

5. Reaction Force:
   According to Newton’s Third Law of Motion, for every action force, there is an equal and opposite reaction force. When one object exerts a force on another, the second object exerts a force of equal magnitude but in the opposite direction on the first object.

6. Linear Momentum:
   Linear momentum is a measure of the motion of an object and is calculated as the product of its mass and velocity. It is a vector quantity, with direction the same as the velocity. The SI unit is kilogram meters per second (kg·m/s).

7. Elastic Collision:
   An elastic collision is a collision in which both kinetic energy and momentum are conserved. In such collisions, the objects involved bounce off each other without any loss of total kinetic energy.

8. Inelastic Collision:
   An inelastic collision is a collision in which momentum is conserved, but kinetic energy is not. In such collisions, some kinetic energy is transformed into other forms of energy, such as heat or sound. In a perfectly inelastic collision, the objects stick together after the collision.

9. Coefficient of Restitution:
   The coefficient of restitution is a measure of the elasticity of a collision between two objects. It is defined as the ratio of the relative speed after the collision to the relative speed before the collision. It ranges from 0 to 1, where:

   • $ 1 $: Perfectly elastic collision (no energy loss).
   • $ 0 $: Perfectly inelastic collision (maximum energy loss, objects stick together).

2.

A.

1. First Law of Motion (Law of Inertia) An object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. This law describes the concept of inertia, which is the tendency of an object to resist changes in its state of motion.

Sports Example:

• In soccer, when a player kicks a stationary ball, the ball remains at rest until the force of the kick is applied. Once the ball is in motion, it would continue moving indefinitely if not for external forces like friction, air resistance, or the goalkeeper stopping it.
• Similarly, in ice hockey, a puck sliding on the ice will continue moving in a straight line at a constant speed unless a player hits it or friction slows it down.

2. Second Law of Motion (F = ma) The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, $ F = ma $, where $ F $ is force, $ m $ is mass, and $ a $ is acceleration. This law explains how the force applied to an object affects its motion, depending on its mass.

Sports Example:

• In baseball, when a batter hits a ball, the force of the bat determines the ball’s acceleration. A harder swing (greater force) results in a faster acceleration of the ball. Conversely, a heavier bat (greater mass) would require more force to achieve the same acceleration.
• In track and field, a shot putter applies a force to the shot put. The greater the force applied, the farther the shot put will travel, but the mass of the shot put also plays a role in determining its acceleration.

3. Third Law of Motion (Action-Reaction) For every action, there is an equal and opposite reaction. When one object exerts a force on another, the second object exerts a force of equal magnitude but in the opposite direction on the first object.

Sports Example:

• In basketball, when a player jumps to dunk the ball, they push down on the ground (action), and the ground pushes them upward with an equal force (reaction), allowing them to leap into the air.
• In swimming, a swimmer pushes backward against the water with their arms and legs (action), and the water pushes them forward with an equal force (reaction), propelling them through the water.

B.

1. Conservation of Momentum In a closed system (where no external forces act), the total momentum before an event (like a collision) is equal to the total momentum after the event. Mathematically, $ m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 $, where $ m $ is mass, $ u $ is initial velocity, and $ v $ is final velocity. Momentum is conserved in collisions or interactions between objects, meaning the total momentum remains constant unless an external force acts on the system.

Example in Sports:

• In billiards, when the cue ball strikes another ball, the total momentum of the two balls before the collision equals the total momentum after the collision. If the cue ball stops after the collision, the other ball moves with the same momentum the cue ball had initially.

2. Impulse-Momentum Relationship The change in momentum of an object is equal to the impulse applied to it. Mathematically, $ \Delta p = F \cdot \Delta t $, where $ \Delta p $ is the change in momentum, $ F $ is the force applied, and $ \Delta t $ is the time over which the force acts. Impulse is the product of force and time, and it directly affects an object’s momentum. A larger force or a longer time of application results in a greater change in momentum.

Example in Sports:

• In baseball, when a batter hits a ball, the force of the bat acting on the ball over a short time interval (impulse) changes the ball’s momentum, causing it to accelerate and fly through the air. A harder swing (greater force) or a longer contact time (follow-through) increases the impulse and the ball’s momentum.

3.

1. $ \sum F = m \cdot a $

Newton’s Second Law of Motion:
The net force ($ \sum F $) acting on an object is equal to the mass ($ m $) of the object multiplied by its acceleration ($ a $). This equation shows how force causes an object to accelerate.

2. $ L = m \cdot v $

Linear Momentum:
The momentum ($ L $) of an object is equal to its mass ($ m $) multiplied by its velocity ($ v $). Momentum is a measure of the motion of an object and is a vector quantity (has both magnitude and direction).

3. $ \sum m_i u_i = \sum m_i v_i $

Conservation of Momentum:
The total momentum of a system before an event (like a collision) is equal to the total momentum after the event. Here, $ m_i $ is the mass of an object, $ u_i $ is its initial velocity, and $ v_i $ is its final velocity.

4. $ m_1 u_1 + m_2 u_2 = (m_1 + m_2) v $

Perfectly Inelastic Collision.
In a perfectly inelastic collision, two objects stick together after colliding. The total momentum before the collision ($ m_1 u_1 + m_2 u_2 $) equals the total momentum after the collision ($ (m_1 + m_2) v $), where $ v $ is the final velocity of the combined objects.

5. $ e = \frac{v_2 - v_1}{u_1 - u_2} $

Coefficient of Restitution:
The coefficient of restitution ($ e $) measures the elasticity of a collision. It is the ratio of the relative speed after the collision ($ v_2 - v_1 $) to the relative speed before the collision ($ u_1 - u_2 $). It ranges from 0 (perfectly inelastic) to 1 (perfectly elastic).

6. $ F \cdot \Delta t = m \cdot \Delta v $

Impulse-Momentum Theorem:
The impulse ($ F \cdot \Delta t $) applied to an object is equal to the change in its momentum ($ m \cdot \Delta v $). Impulse is the product of force ($ F $) and the time ($ \Delta t $) over which it acts.