Laws of Motion - Exhaustive Study Guide and Self-Test Analysis

Subject: Science and Technology Digest : STD. IX Self-Test.
Chapter: 1. Laws of Motion.
Allocated Time: 1 Hour.
Maximum Marks: 20 Marks.
Administrative Note: Answers are meant to be verified against the official model answers provided via an external link.

Definition of an Object at Rest: A body is said to be at rest when it does not change its position with respect to its surrounding environment over a given period of time.
Acceleration Units: In the Standard International (SI) system, the unit for acceleration is meters per second squared, expressed as $\text{m/s}^2$ .
Direction of Momentum: The change in momentum of an object occurs explicitly in the direction of the applied force.
Momentum Formula: Expressed in usual notation, $\text{Momentum} = \text{mass} \times \text{velocity}$ ( $p = m \times v$ ).
- It is distinct from Kinetic Energy ( $\frac{1}{2}mv^2$ ) and Potential Energy ( $mgh$ ).
- It is also distinct from the product of mass and speed, as momentum is a vector quantity that requires velocity (direction) rather than just scalar speed.

Odd One Out Identification: In a set containing Momentum, Acceleration, Force, and Time, the odd quantity is Time.
- Reasoning: Momentum, acceleration, and force are all vector quantities, meaning they possess both magnitude and direction. Time is a scalar quantity, defined solely by its magnitude.

Uniform Motion vs. Non-uniform Motion:
- Uniform Motion: A body is said to be in uniform motion if it covers equal distances in equal intervals of time, regardless of how small those time intervals may be.
- Non-uniform Motion: A body is said to be in non-uniform motion if it covers unequal distances in equal intervals of time. This occurs when the object's velocity changes, whether in magnitude or direction.

Simple Momentum Calculation:
- Problem: A body of mass $10\,kg$ moves with a velocity of $5\,m/s$ . Find its momentum.
- Formula: $p = m \times v$
- Calculation: $p = 10\,kg \times 5\,m/s = 50\,kg\,m/s$
Conservation of Momentum (Collision and Embedding):
- Scenario: A bullet of mass $10\,g$ ( $0.01\,kg$ ) moving at $1.5\,m/s$ penetrates a wooden plank of mass $90\,g$ ( $0.09\,kg$ ). The plank is initially at rest.
- Law applied: The Law of Conservation of Momentum states that the total momentum before the impact equals the total momentum after the impact.
- Initial Momentum Calculation:
  - Bullet momentum ( $m_1u_1$ ) = $0.01\,kg \times 1.5\,m/s = 0.015\,kg\,m/s$ .
  - Plank momentum ( $m_2u_2$ ) = $0.09\,kg \times 0\,m/s = 0\,kg\,m/s$ .
  - Total initial momentum = $0.015\,kg\,m/s$ .
- Final Momentum Calculation:
  - The bullet gets embedded, so the combined mass is $m_1 + m_2 = 0.01\,kg + 0.09\,kg = 0.1\,kg$ .
  - Let the common velocity be $V$ .
  - Equation: $0.015 = 0.1 \times V$
  - Velocity ( $V$ ) = $\frac{0.015}{0.1} = 0.15\,m/s$ .

Positive Acceleration:
- Definition: When the velocity of an object increases with time, the acceleration is considered positive.
- Example: A car starting from a stationary position and accelerating on a straight highway to reach a higher speed.
Negative Acceleration (Deceleration or Retardation):
- Definition: When the velocity of an object decreases with time, the acceleration is negative.
- Example: A cyclist applying the brakes to come to a stop, causing the speed of the bicycle to decrease.
Zero Acceleration:
- Definition: If the velocity of the object remains constant over time (no change in magnitude or direction), the acceleration is zero.
- Example: A car traveling on a straight road using cruise control at a constant speed of $80\,km/h$ .

Core Principle: For every action force, there is an equal and opposite reaction force that acts simultaneously.
Examples:
1. Recoil of a Gun: When a bullet is fired (action), the gun exerts a forward force on the bullet. Simultaneously, the bullet exerts an equal and opposite backward force on the gun, causing it to recoil.
2. Rocket Propulsion: A rocket ejects high-velocity exhaust gases downward (action). The gases exert an equal upward force on the rocket (reaction), propelling it into space.
3. Swimming: A swimmer pushes the water backward with their hands and feet (action). The water pushes the swimmer forward with an equal and opposite force (reaction).
4. Walking: To walk, a person pushes the ground backward with their feet (action). The ground exerts an equal and opposite force pushing the person forward (reaction).
5. Rowing a Boat: A rower pushes the water backward with the oars (action). The water reacts by pushing the boat forward (reaction).