2025.02

Homework Format

• Expectation for homework submission:

  • Homework must be presented vertically, step-by-step.

  • Avoid submitting scratch paper; explanations must be clear.

Ballistic Motion

• Important Principles:

  • Do not mix x and y components of motion: acceleration, velocity, and displacement.

• Projectile Motion:

  • The path is called a "trajectory".

  • In the absence of air resistance, projectile paths are parabolic.

  • All objects, regardless of their trajectories, will hit the ground at the same time if launched simultaneously from the same height.

Paths of Objects in Gravitational Fields

• Attributes of trajectories:

  • Near Earth's surface: pathways form parabolas.

  • For objects in orbit: possible paths include circles, ellipses, parabolas, and hyperbolas.

Newton’s Cannon Analysis

• Setup:

  • 5.00 m high cliff.

• Calculations:

  • Given distance (d), find horizontal velocity (vx).

  • Given horizontal velocity (vx), find distance (d).

  • Example calculations:

    • Let d = 100.0 m,

    • Let v = 600.0 m/s horizontally.

• Method: Treat x and y motions separately to determine values.

Components of Velocity

• Initial Velocity:

  • If the initial velocity is not purely horizontal, separate into x and y components.

• Ballistic Motion Strategies:

  • Regularly separate initial velocity into x and y components for analysis.

  • It is essential to solve for time, even if not required explicitly.

Motion Behavior in Ballistic motion

• Vertical velocity at the peak of the trajectory is zero.

• Velocity just before impact is not zero.

• Acceleration Details:

  • Vertical acceleration is constant at 9.80 m/s² downward.

  • Horizontal acceleration is zero, maintaining constant horizontal velocity.

• Critical Understanding:

  • Treat horizontal and vertical motions separately.

  • Vertical motion reveals time until hitting the ground, whereas horizontal motion remains independent until contact with another object.

Projectile Analysis: Golf Ball Example

• Questions about Velocity:

  • What happens to horizontal velocity on ascent?

    • Options include: A. Bigger, B. Smaller, C. Nothing, D. Depends on the angle.

  • What happens to vertical velocity on ascent?

    • Asks similar options.

• High Point Direction:

  • Questions about the direction at the peak of the trajectory—options include vertical and angled trajectories.

• Angle Launch:

  • For a projectile launched at 30°: asks about acceleration throughout the trajectory (not varying).

Example Projectile Motion Calculations

• Calculation Details:

  • A ball thrown with initial velocities (vy = 9.8 m/s and vx = 30 m/s) asks for:

    • Time to rise and return to height,

    • Horizontal distance traveled in that time.

• Derived formula: [ d_x = \frac{2v_i^2 \sin(\theta)\cos(\theta)}{g} ]

  • Maximum distance occurs at an angle of 45°.

Demonstration Example

• Launch Scenarios:

  • Golf ball hit at different angles (30°, 60°) and initial speeds (50 m/s), comparing distances, heights, and impact times.

  • Key conclusions drawn about the distance traveled and maximum height attained by projectiles at specific angles.

Uniform Circular Motion

• Characteristics of Circular Motion:

  • Centripetal acceleration is always perpendicular to velocity; the speed remains unchanged while direction changes constantly.

• Acceleration and Force Relationships:

  • Acceleration and force will alter the direction and speed of an object depending on their relative directions.

• Important Equations:

  • ( a_c = \frac{v^2}{r} )

  • ( F_c = ma_c = \frac{mv^2}{r} )

• Centripetal vs. Centrifugal: Centripetal force directs toward rotation's center; centrifugal appears in accelerated frames only.

Example Problems

• Compute centripetal acceleration and force for given scenarios (e.g., a ball at a radius with specific velocity).

• Frame questions regarding forces experienced during circular motion, such as identifying the force direction when acceleration occurs.