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Recording-2025-02-27T16:01:26.422Z

Conic Sections and Trajectories

  • Conic Sections: Shapes formed by the intersection of a plane and a cone; types include circles, ellipses, parabolas, and hyperbolas.

    • Circle: Cross-section cut straight across the cone.

    • Ellipse: Cross-section cut diagonally; an oval shape defined by a specific mathematical formula.

    • Hyperbola: Created when the cone is cut more vertically.

  • Parabolic Trajectories: When gravity is the only force acting, objects follow a parabolic path, especially when near Earth's surface.

Motion Under Gravity

  • Objects in free fall near Earth experience constant acceleration due to gravity, approximately 9.8 m/s² downward.

Vertical Motion Analysis

  • Fall Height: For a drop from a height of 5 meters:

    • Initial vertical velocity = 0 (as it's shot horizontally).

    • The positive direction can be set downwards for simplicity.

    • Equation to find time of fall using one-dimensional motion:[ d = \frac{1}{2} a t^2 + v_{initial} t ]Rearranging:[ d_y = \frac{1}{2} (9.8) t^2 ]

    • Setting parameters: Falling distance (d_y) = 5m.

    • Solving gives: [ t \approx 1.01 \text{ seconds}. ]

  • Distance Traveled Horizontally: For a horizontal distance of 100 meters:

    • Using time found above to calculate horizontal distance via [ d_x = v_{initial} \times t ]

    • Using [ d_x = 100 m ]: [ 100 = 30 \times 1.01 ]

      • Concludes with horizontal distance calculation for 30m/s in the horizontal direction.

Vertical Velocity Dynamics

  • Trajectory Characteristics:

    • At the peak, vertical velocity = 0 m/s; returning to starting height results in the same speed but opposite direction.

    • Vertical acceleration remains constant (9.8 m/s² downward) for the duration.

    • If thrown upwards, as it ascends, velocity decreases until it reaches zero; then descends, accelerating due to gravity.

Horizontal Motion

  • Acceleration: In the x-direction, acceleration = 0 m/s²; horizontal velocity remains constant throughout the trajectory.

  • Displacement:

    • Total vertical displacement on return to the same height = 0 m.

    • For the trip: A calculation for distances and times needed horizontally and vertically.

Additional Projectile Motion Calculations

  • Calculating Time of Flight:

    • For vertical displacement under acceleration: [ d_{y} = v_{initial,y} \times t + \frac{1}{2} a_{y} t^2 ]

    • Adjusted for vertical velocity direction (up positive, down negative): [ d_{y} ] results in zero if returning to horizontal displacement.

Angled Projectile Motion

  • For a projectile launched at an angle (ex. golf ball at 45 degrees):

    • Horizontal and vertical components separated:

      • [ v_{x} = v_{initial} \cos(theta) ]

      • [ v_{y} = v_{initial} \sin(theta) ]

    • Time to peak flight can be calculated, followed by total horizontal distance using horizontal motion equations.

    • Final distance also calculated incorporating angle calculations, sine, and cosine components.

Circular Motion and Acceleration

  • Uniform Circular Motion:

    • Despite speed not changing while moving in a circle, direction does, leading to acceleration.

    • Acceleration vector is perpendicular to the velocity vector, directing towards the center of the circle (centripetal acceleration).

    • Understanding of directional components of velocity/acceleration crucial for complex motion calculations.