(455) Motion in 2D - projectile motion [IB Physics SL/HL]

Projectile Motion

• Projectile motion involves two-dimensional movement, which can be visualized as a trajectory that goes up and then down, resembling a parabola.

• Understanding this motion requires breaking down the velocity into its components, the horizontal (x-direction) and vertical (y-direction).

Velocity Components

• A velocity vector (V) at an angle θ can be split into:

  • Horizontal Component (Vx): Vx = V cos(θ)

  • Vertical Component (Vy): Vy = V sin(θ)

• Vx represents constant speed (no acceleration) in horizontal motion.

• Vy is affected by gravity, leading to acceleration in vertical motion (downward).

Motion Equations

• Horizontal Motion: Constant velocity, so: Vx = displacement (Sx) / time (T).

• Vertical Motion:

  • Uses equations of accelerated motion.

  • At maximum height, vertical velocity (Vy) becomes 0, while horizontal velocity (Vx) remains constant.

Example Problem: Launch from a Cliff

• Scenario: Object launched from a 100m high cliff at 50 m/s at a 30° angle.

  • Break down initial velocity:

    • Vx = 50 cos(30°) = 43.3 m/s

    • Vy = 50 sin(30°) = 25 m/s

• Maximum Height Calculation:

  • At max height, vertical speed is 0.

  • Using V² = U² + 2aS, where V=0, U=25 (initial vertical speed), a=-9.81 (gravity), solve for S:

    • S = -(U²)/(2a) = -(25²)/(2 * -9.81) = 31.855m

  • Total maximum height from ground = cliff height (100m) + vertical height (32m) = 132m.

Horizontal Displacement Calculation

• Find horizontal displacement (Sx) during flight:

  • Use Sx = Vx * T, where T is the total time in the air.

• Vertical Motion: Final displacement is 100m down:

  • S = Ut + ½at² yields:

  • 100 = 25T + 4.95T², leading to a quadratic equation to solve for T.

• After calculating, T ≈ 7.7 seconds.

• Final horizontal distance: Sx = Vx * T = 43.3 m/s * 7.7 s = 333m (or 330m rounded).

Conclusion

• Understanding the separation of components in projectile motion allows for methodical calculations of maximum height and horizontal distance.