Kinematics-Projectiles and Free Fall

Introduction to Projectile Motion

  • A projectile is an object that is thrown into the air with an initial speed and moves under the influence of Earth's gravity.
  • The path followed by a projectile is called its trajectory, which follows a parabolic shape neglecting air resistance.
  • The motion of a projectile can be broken down into horizontal (x) and vertical (y) components that are mathematically independent.

Basic Concepts

  • Horizontal Component:
    • Remains constant (a_x = 0) throughout the flight if we neglect air resistance.
    • The object covers equal horizontal distances in equal time intervals.
  • Vertical Component:
    • Experiences constant acceleration due to gravity (g \approx 9.81 \text{ m/s}^2 downward).
    • The vertical velocity decreases as the object rises, becomes zero at the peak, and increases as it falls.

Vector Representation of Projectile Motion

Example 1: Volleyball Hit at an Angle
  • A volleyball hit into the air has an initial speed of 10 \text{ m/s}.
  • The optimal angle to maximize time in the air is 90^\circ (fired straight up).
  • Vectors for varying angles:
    • 90^\circ: Maximum time of flight, zero horizontal range.
    • 45^\circ: Maximum horizontal range (on level ground).
    • 30^\circ and 60^\circ: Complementary angles that result in the same horizontal range but different flight times and maximum heights.
Example 2: Initial Velocity Components
  • A projectile fired from the ground has an initial velocity (v_0) of 250 \text{ m/s} at an angle of 60^\circ above horizontal.
  • Calculation of Components:
    • Horizontal Component (V_x):
    • Vx = v0 \cos(\theta) = 250 \cos(60^\circ) = 125 \text{ m/s}
    • Vertical Component (V_y):
    • Vy = v0 \sin(\theta) = 250 \sin(60^\circ) \approx 216.5 \text{ m/s}

Trajectory Characteristics and Symmetry

  • Maximum Height (H): Occurs when the vertical velocity component (V_y) reaches 0 \text{ m/s}.
  • Symmetry of Flight: For a projectile launched and landing at the same height:
    • The time taken to reach maximum height is exactly half of the total flight time (t{\text{up}} = t{\text{down}}).
    • The impact speed is equal to the launch speed, and the impact angle is the negative of the launch angle.
  • Velocity Vector: At any point, the velocity vector is tangent to the path. The horizontal component stays at 125 \text{ m/s}, while the vertical component changes.

Projectile Motion Questions & Solutions

Question 5: Comparing Two Stones
  • Two stones (Stone A: 15 \text{ m/s}, Stone B: 30 \text{ m/s}) are thrown horizontally from a cliff.
  • Time to reach ground: Both take the same time to reach the ground because vertical motion is independent of horizontal speed. Since they start with the same initial vertical velocity (0 \text{ m/s}) and fall the same distance, gravity acts on them equally.
Path of a Stunt Car Off a Cliff
  • Horizontal velocity remains constant (neglecting friction) throughout the fall.
  • Vertical velocity increases linearly with time (v_y = gt).
  • The resultant velocity magnitude is calculated using the Pythagorean theorem: v = \sqrt{Vx^2 + Vy^2}.

Vertical Motion Equations

  1. Final Velocity: vy = uy + a_y t
  2. Displacement: dy = uy t + \frac{1}{2} a_y t^2
  3. Velocity-Squared: vy^2 = uy^2 + 2 ay dy
  • Note: Use a_y = -9.81 \text{ m/s}^2 if defining upward as the positive direction.

Important Formulas for Level Ground

  • Horizontal Range (R):
    • R = \frac{v_0^2 \sin(2\theta)}{g}
  • Time of Flight (T):
    • T = \frac{2 v_0 \sin\theta}{g}
  • Maximum Height (H_{max}):
    • H{max} = \frac{(v0 \sin\theta)^2}{2g}

Overview of Air Resistance

  • In theoretical problems, air resistance is often neglected to simplify motion to a perfect parabola.
  • In Real-World Applications:
    • Air resistance (drag) depends on the object's speed, shape, and cross-sectional area.
    • Drag reduces both the maximum height and the horizontal range.
    • The trajectory becomes asymmetrical, with the descent being steeper than the ascent.

Common Misconceptions

  • Misconception: Heavier objects fall faster in projectile motion.
    • Reality: Acceleration due to gravity is independent of mass (g is constant for all objects).
  • Misconception: There is a horizontal force keeping the object moving.
    • Reality: Once launched, gravity is the only force (neglecting air resistance). Inertia maintains the horizontal velocity.

Conclusion

  • Mastery of kinematics involves isolating the constant-velocity horizontal motion from the constant-acceleration vertical motion and linking them using the common variable of time (t).