Motion of objects

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