Circular Motion: Distance, Displacement, Velocity

Distance and Displacement in Circular Motion

• Total distance from point A to point B in circular motion equates to one-quarter of the circumference if the path is a quarter rotation.
• A quarter rotation and a three-quarter rotation result in equal displacement magnitudes.
• For a full circle, the total angle is 2π radians.
• The total circumference of a circle is 2πr.

Resultant Vector and Velocity Direction

• The resultant vector lies in the fourth quadrant, indicating an East-South direction.
• Velocity direction aligns with displacement direction, pointing Southeast.

Velocity and Time Relationship

• When equal distances are covered in both directions, different velocities indicate varying travel times.
• A higher velocity on the first journey implies a shorter time (T1) compared to the return journey (T2).
• If time taken is the same for paths A to B and B to C, differing velocities mean different distances.
• Higher velocity from A to B suggests a greater distance compared to B to C, where velocity is lower.

Acceleration and Graphical Representation

• Acceleration direction aligns with the direction of changing velocity and applied force.
• Throwing a body upward correlates to a negative acceleration of -9.8 m/s².
• A falling body experiences positive acceleration of +9.8 m/s².
• Uniform acceleration means velocity changes consistently by 9.8 m/s every second.

Velocity-Time Graph and Acceleration

• The slope of a velocity-time curve signifies instantaneous acceleration.
• Acceleration varies at every point along a non-linear velocity-time graph.
• The slope of a graph does not always equate to a tangible physical quantity.
• The slope is equivalent to the gradient.

Tangent and Slope

• Higher values of tan θ relate to steeper slopes.
• Larger θ indicates a greater inclination relative to the X-axis.
• As θ increases from 0° to 90°, the slope also increases.
• At 90°, the slope approaches infinity.
• Slope is positive in the first and third quadrants.
• Slope is negative in the second and fourth quadrants.
• An angle less than 90° indicates a positive slope.
• An angle greater than 90° indicates a negative slope.
• The area of a graph is calculated as Height × Base or Perpendicular × Width.
• A trapezium has two parallel sides and two unequal, non-parallel sides.

Graph Area and Motion Relationships

• The area under an acceleration-time graph represents the change in velocity.
• The area under a velocity-time graph represents displacement.
• On a distance-time graph, a line parallel to the X-axis indicates:
  • An angle of 0° with the X-axis.
  • tan θ = 0.
  • Velocity = 0.
• Variable velocity (positive) on a distance-time graph shows increasing velocity.
• Decreasing velocity is represented by negative velocity.
• Increasing deceleration can also be graphically represented.

Slope and Theta Angle Relationship

• As θ approaches 90°, the slope increases.
• As θ approaches 180°, the slope decreases.
• In a specific example, an initial slope angle of 180° moving toward 90° decreases θ.
• Despite decreasing θ, the slope increases but remains negative.
• Velocity is negative but increasing, showing that increasing velocity represents increasing acceleration.

Graph Slope Interpretation

• The slope at any point on a graph equals acceleration at that time.
• The area under the graph indicates total displacement.

Graph (a) Interpretation

• An object begins with positive velocity.
• The object slows down over time.
• The area under the curve is negative, indicating displacement in the negative direction.

Graph (b) Interpretation

• An object starts with positive velocity.
• The object decelerates.
• The object reverses direction.
• The area under the curve is negative, showing displacement in the negative direction.

Projectile Motion and Acceleration

• When going up, gravity slows the projectile, resulting in negative acceleration.
• At the top of its path, gravity continues to act, preparing to bring it down.
• When coming down, gravity accelerates the projectile, but acceleration remains negative.
• The initial direction was upward, and gravity acts downward from the start.

Velocity-Time Graph Segments

• Segment A to B, a horizontal line, indicates constant velocity with no acceleration.
• Segment B to a lower point, a vertical drop, indicates an instantaneous velocity change.
• It drops to a lower velocity, potentially zero, and represents rapid velocity change in idealized conditions.
• Segment C to a lower point, another vertical drop, indicates an instantaneous velocity decrease.
• It is also an idealized representation of rapid velocity change.

Key Concepts in Velocity-Time Graphs

• Constant velocity is represented by horizontal lines, indicating zero acceleration.
• Instantaneous change in velocity is represented by vertical drops, an idealized concept.
• Velocity changes in the real world occur over brief time periods, not instantaneously.

Inertial and Non-Inertial Frames of Reference

• An inertial frame of reference has zero acceleration (a = 0).
• An object at rest remains at rest, and an object in motion maintains constant velocity, as per Newton’s First Law.
• Without external force, motion remains unaltered.
• A non-inertial frame of reference is an accelerating reference frame, such as a rotating carousel.
• Newton’s laws become more complex to apply in such frames.

Force and Time Relationship (Impulse)

• Force is inversely proportional to the time interval (F ∝ 1/Δt).
• Decreasing force leads to an increased time interval.
• Increasing force results in a decreased time interval.
• In car crashes, airbags increase the impact time (Δt).
• This reduces the force experienced by occupants for the same change in momentum.

Momentum (P) and Impulse Relationship

• Momentum (P) is the product of mass and velocity (P = mv).
• Impulse (force × time) induces a change in momentum.
• Applying force over time causes an object’s momentum to change by the impulse amount.
• The direction of impulse and momentum change always corresponds to the direction of the applied force.

Wooden Block and Firecracker Explosion

• Initially, a wooden block is at rest.
• A firecracker is inside the block with negligible mass (approximately zero).
• Upon explosion:
  • Cracker fragments/ejected material: 50g.
  • Wooden block: 20g.
  • Velocity of cracker fragments: 100 cm/s.
• A ball thrown upward slows, stops at its max height, then accelerates downward.

Balloon Momentum Principle

• The momentum of the balloon is equal to the momentum of the air ejected from it but in the opposite direction.

Acceleration and Graphs

• Determine the fraction of distance covered with positive acceleration by dividing the corresponding area by total displacement.
• Determine zero acceleration similarly using the area under the velocity-time graph.

Spring and Attached Bodies

• Release of two bodies attached to a compressed spring results in motion where one body moves right and the other moves left.

Explosion of a Can with a Wooden Block

• The cracker's mass is negligible.
• The mass of the can is 50g.
• The mass of the wooden block closing the can is 20g.
• Upon explosion, the can and wooden block move in opposite directions.

Relative Velocity

• Same direction: Subtract velocities.
• Opposite directions: Add velocities.
• Application: Doppler effect.

Elastic Collision Example

• A heavy steel ball (5 kg) collides with a tennis ball (0.2 kg) at rest.
• The steel ball’s velocity remains almost unchanged.
• The tennis ball’s velocity becomes approximately twice that of the steel ball’s initial velocity.

Projectile Motion Principles

• Horizontal acceleration is zero.
• Vertical acceleration is -g.
• Use the Pythagorean theorem to find total acceleration by squaring values to remove negative signs, thereby making total acceleration positive.
• Upward direction: Positive.
• Downward direction: Negative (-g).

Hyperbolic Path

• Requires infinite motion in both directions.
• Physically impossible.

Parabolas vs. Hyperbolas

• Hyperbolas:
  • Exist in both positive and negative quadrants.
• Parabolas:
  • U-shaped.
  • Exist in either positive or negative quadrants, but not both.

Angle and Range in Projectile Motion

• Complementary angles (summing to 90 degrees) yield equal ranges.
• Maximum range is achieved at an angle closest to 45 degrees.
• Horizontal velocity remains constant in projectile motion.
• Initial and final velocities are interchangeable in formulas.

Equations of Motion Usage

• The second equation of motion is frequently used for projectile motion calculations.

Maximum Height in Projectile Motion

• Theoretical Maximum:
  • Occurs at 90 degrees, but is a special case.
• Practical Consideration:
  • For maximum range, use the angle closest to 45 degrees.
  • For maximum height, use the angle closest to 90 degrees.

Time to Reach Maximum Height

• It is half of the total time of flight because the time ascending equals the time descending.

Approximate Square Root Calculation

• The value is replaced with a nearby number whose square root is known.
• For example, approximate √320 using √400 = 20.

Types of Projectile Motion

• Oblique Projectile Motion:
  • A body is launched at an angle.
• Horizontal Projectile Motion:
  • Body is dropped from a height.
  • Initial height is nonzero.
  • Launch angle is 0 degrees, so the Y-component of initial velocity is zero, forming a half-parabola.

Vertical Projectile Motion

• Range is zero.

Gravity’s Role in Horizontal Projectile Motion

• Direction of Gravity:
  • Taken as positive, with gravitational acceleration downward.
• Vertical Component of Velocity:
  • Increases in the direction of gravity, causing the body to accelerate downward.

Relative Motion in Projectile Scenarios

• If person A throws a ball and person B moves in the opposite direction:
  • The ball follows a parabolic path.
  • Their relative velocity decreases.

Motion of a Dropped Object from a Moving Car

• If a ball is dropped:
  • Inherits the car’s horizontal velocity.
  • Does not land at the dropping point but follows a half-parabolic trajectory.
• If a ball is thrown upward:
  • The motion is parabolic, combining initial upward velocity with downward gravitational acceleration.

Simultaneous Falling Objects

• A dropped ball and a horizontally thrown ball fall at the same rate and hit the ground simultaneously, as horizontal motion does not affect vertical motion.
• Both experience identical gravitational acceleration and have an initial vertical velocity of zero.

Displacement vs. Speed Graph

• A body thrown upward stops at its highest point which dictates the start point of the graph, if based on this point.

Teacher’s Error in Understanding Motion

• A decreasing velocity mislabeled by a teacher as non-uniform should be increasing non-uniform velocity in the opposite direction.