Motion of objects Questions

Correct Answer

📖 Governing Law / Core Concept

Motion under uniform acceleration is governed by the equations of kinematics, specifically:

where:

• v = final velocity (m/s)
• u = initial velocity (m/s)
• a = acceleration (m/s²)
• t = time (s)

🧪 Step-by-Step Breakdown

Given:

• Initial velocity, \( u = 40 \, \text{m/s} \)
• Final velocity at maximum height, \( v = 0 \, \text{m/s} \)
• Acceleration due to gravity, \( a = -10 \, \text{m/s}^2 \)

Applying the kinematic equation:

Rearranging gives:

The ball reaches a maximum height in approximately 4 seconds.

Correct Answer

📖 Governing Law / Core Concept

The concept involved in this scenario is based on Newton's First Law of Motion, which states that an object in motion will remain in motion unless acted upon by an external force. Circular motion implies a constant speed but a continuously changing velocity due to the change in direction.

🧪 Step-by-Step Breakdown

A car moving in uniform circular motion maintains a constant speed; however, its direction is constantly changing. The key factors involved include:

• Velocity (\( v \)): Change in both magnitude and direction.
• Acceleration (\( a \)): Centripetal acceleration required to keep the car moving in a circle, given by:
  \[ a = \frac{v^2}{r} \]
  where \( r \) is the radius of the circular path.
• Momentum (\( p \)): Defined as \( p = mv \), where \( m \) is the mass of the car. Since the direction of the velocity vector changes, the momentum changes continuously as well.

🔍 Option Analysis

Option (a): Incorrect; speed remains constant, but direction changes.
Option (c): Incorrect; a change does occur in both direction and speed.
Option (d): Incorrect; momentum cannot remain constant if velocity is changing.

⚡ Mnemonic / Speed-Run

Remember: "Centripetal means center-seeking," which reminds that centripetal force is what keeps an object in a circular path, thereby influencing its velocity and momentum even when speed is uniform.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The relevant formula governing the motion of an upward-moving object is derived from the equations of motion under uniform acceleration due to gravity. Specifically, the maximum height \(h\) attained can be given by: \[ h = \frac{v^2}{2g} \] where \(v\) is the initial velocity and \(g\) is the acceleration due to gravity (approximately \(9.81 \, \text{m/s}^2\)).

🧪 Step-by-Step Breakdown

Given that the maximum height \(h = 20 \, \text{m}\): \[ 20 = \frac{v^2}{2 \times 9.81} \] Rearranging gives: \[ v^2 = 20 \times 2 \times 9.81 = 392.4 \] Taking the square root: \[ v = \sqrt{392.4} \approx 19.8 \, \text{m/s} \] Thus, \(v\) is approximately \(20 \, \text{m/s}\).

🔍 Option Analysis

Options A (8 m/s), B (12 m/s), and C (16 m/s) are incorrect as they would not provide enough initial velocity to reach the maximum height of 20 m given the gravitational acceleration.

⚡ Mnemonic / Speed-Run

Remember \(h = \frac{v^2}{2g}\) as "Height equals velocity squared over twice gravity." This formula summarizes the energy conversion from kinetic to potential energy.

6. Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The speed \( V \) of an object is defined as the magnitude of its velocity, which can be calculated from the slope of the position vs. time graph:

🧪 Step-by-Step Breakdown

From the provided graph:

Hence, the relation is:

🔍 Option Analysis

• Option (a): Incorrect, as Vₐ is not less than Vᵦ.
• Option (c): Incorrect, all speeds are not zero as indicated by positive slopes.
• Option (d): Incorrect, since their speeds are unequal.

⚡ Mnemonic / Speed-Run

Remember: Steep slopes indicate higher speeds. Graphically visualize it as a race — the steeper the hill, the faster the runner.

📖 Factual Verification & Reference

Correct Answer

📖 Core Concept

The relationship between velocity (V) and time (t) is crucial in kinematics. A velocity-time graph illustrates the motion of an object, showing periods of acceleration (positive slope) and deceleration (negative slope).

🧪 Step-by-Step Breakdown

• Segment AB: Represents acceleration. Here, velocity increases as time progresses.
• Segment BC: Represents uniform motion, where velocity remains constant.
• Segment CD: Represents deceleration, where velocity decreases.

🔍 Option Analysis

• Option A: CD and BC - Incorrect; BC indicates constant velocity.
• Option B: BC and AB - Incorrect; BC does not denote deceleration.
• Option D: AB and CD - Incorrect; only CD indicates deceleration.

⚡ Mnemonic / Speed-Run

Remember: "Accelerate Upwards, Decelerate Downwards" for segment identification.

📊 Visual Suggestion

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The principle of conservation of momentum applies here: an internal process (chemical reaction) cannot affect the overall momentum of an isolated system.

🧪 Step-by-Step Breakdown

Consider a system in motion with uniform velocity \( v \). The center of mass of this system is given by:

Where:

• \( m_i \) = mass of each particle
• \( v_i \) = velocity of each particle

Since the total external force acting on the system is zero during a uniform chemical reaction, the center of mass velocity \( v_{cm} \) remains constant.

Regarding kinetic energy, while internal energy states change, the system's overall kinetic energy with respect to its center of mass does not change due to the conservation laws.

🔍 Option Analysis

• (b) 2 only: Incorrect, as kinetic energy changes are internal to the system.
• (c) Both 1 and 2: Incorrect, since 2 is false.
• (d) Neither 1 nor 2: Incorrect, as 1 is valid.

⚡ Mnemonic / Speed-Run

Remember: "Internal changes do not affect the external motion." This ties into the conservation principles.

📖 Factual Verification & Reference

Correct Answer

📖 Governing Law / Core Concept

The equation of motion that governs the relationship between distance \( s \), initial velocity \( u \), acceleration \( a \), and time \( t \) is given by:

🧪 Step-by-Step Breakdown

From the equation of motion:

This shows that:

• The term \( ut \) indicates that the distance \( s \) is directly influenced by the initial velocity \( u \).
• The term \( \frac{1}{2} a t^2 \) contributes equally to the distance based on acceleration and time but does not negate the dependency on the initial velocity.

Thus, the total distance covered incorporates the initial velocity directly.

🔍 Option Analysis

• Option (b): is independent of its initial velocity - Incorrect, as shown by the \( ut \) term.
• Option (c): increases linearly with time - Incorrect; while distance does change over time, it exhibits a quadratic relationship due to acceleration.
• Option (d): depends on its initial displacement - Incorrect, as displacement is not a factor in the context of this question.

⚡ Mnemonic / Speed-Run

Recall: "Initial speed sets the stage, while acceleration lifts the weight." This aids in remembering how initial velocity significantly affects motion dynamics.

📖 Factual Verification & Reference

Solution

Option (a) is correct.

Explanation:

Solution

Option (c) is correct.

Explanation:

Using the initial velocity \( u = 0 \), acceleration \( a = 2 \, \text{m/s}^2 \), and time \( t = 10 \, \text{s} \): Next, for the second segment towards the south with \( u = 0 \), \( a = 4 \, \text{m/s}^2 \), and \( t = 10 \, \text{s} \): Net displacement is found using the Pythagorean theorem: Substituting \( s_1 \) and \( s_2 \): Thus, the net displacement is \( 300 \, \text{m} \).