Section 1.6: Advanced Kinematics

Advanced kinematics explores motion beyond constant acceleration and simple 2D trajectories. We cover variable acceleration, circular motion, relative motion in 2D, and motion with resistive forces.

1. Variable Acceleration (1D):

For acceleration that changes with time \(a(t)\), velocity and position are found by integration:

  • Velocity: \( v(t) = v_0 + \int a(t) dt \)
  • Position: \( x(t) = x_0 + \int v(t) dt \)
2. Relative Motion in 2D:

Velocity relative to the ground is:

\( \vec{v}_{BG} = \vec{v}_{BW} + \vec{v}_{WG} \)

  • Decompose vectors into x and y components for calculations.

Example: A boat moving at 5 m/s in water flowing at 3 m/s perpendicular.

3. Circular Motion:
  • Centripetal acceleration (toward center): \( a_c = \frac{v^2}{r} \)
  • Tangential acceleration (if speed changes): \( a_t = \frac{dv}{dt} \)
4. Motion with Resistance:

Linear resistive force: \( F_r = kv \)

Equation of motion: \( m \frac{dv}{dt} = F - kv \)

Requires integration or approximation to solve.

Example 1: Variable Acceleration

An object starts from rest with \( a(t) = 4t \) m/s². Find velocity and position after 3 s.

Velocity: \( v = \int_0^3 4t dt = 4 * 9/2 = 18 \text{ m/s} \)

Position: \( x = \int_0^3 v(t) dt = \int_0^3 2 t^2 dt = 18 \text{ m} \)

Example 2: Circular Motion

A car moves at 20 m/s around a circular track of radius 50 m. Find its centripetal acceleration.

\( a_c = \frac{v^2}{r} = \frac{20^2}{50} = 8 \text{ m/s²} \)

Practice Problems

  1. An object accelerates according to \( a(t) = 3t \) m/s² from rest. Find velocity and displacement after 4 s.
  2. A boat moves at 6 m/s in water flowing at 4 m/s perpendicular. Find resultant velocity relative to shore.
  3. A car turns in a circle of radius 40 m at 15 m/s. Find centripetal acceleration.
  4. An object experiences resistive force \( F_r = kv \) with k=2 kg/s and driving force 10 N. Find steady velocity.
  5. A particle moves in 2D with \( v_x = 3 \) m/s, \( v_y = 4 \) m/s. Find speed and direction.
  6. A car accelerates from 0 to 30 m/s in 10 s while moving around a circular track of radius 100 m. Find tangential and centripetal accelerations at the end.
  7. A drone moves with velocity 5 m/s north relative to wind moving 2 m/s east. Find velocity relative to ground.
  8. An object accelerates according to \( a(t) = 6 - t \) m/s² from rest. Find velocity and displacement after 5 s.
  9. A cyclist decelerates at a rate proportional to velocity, \( F_r = kv \), starting at 12 m/s. If k=1 s⁻¹, find velocity after 3 s.
  10. A particle travels along a curved path with radius 10 m and speed 8 m/s. Find centripetal acceleration.
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