Section 1.8: Advanced Problems
This section contains challenging problems combining concepts from kinematics, vectors, derivatives, and projectile motion. Designed to test deep understanding and problem-solving agility.
Example 1
A projectile is fired from a cliff 50 m high with initial speed 30 m/s at 40°. Find total flight time and horizontal distance from base of cliff.
v_x = 30 cos40° ≈ 22.98 m/s
v_y0 = 30 sin40° ≈ 19.28 m/s
Use y = y0 + v_y0*t - 0.5 g t² = 0 → 0 = 50 + 19.28 t - 4.9 t²
Quadratic: 4.9 t² - 19.28 t - 50 = 0 → t ≈ 6.68 s
Horizontal distance: x = v_x * t ≈ 22.98 * 6.68 ≈ 153.5 m
Example 2
Particle moves along x-axis with x(t) = 5t³ – 3t² + 2t. Find acceleration at t = 2 s and velocity when a = 0.
v(t) = dx/dt = 15t² – 6t + 2
a(t) = dv/dt = 30t – 6 → a(2) = 30*2 – 6 = 54 m/s²
Set a(t) = 0 → 30t – 6 = 0 → t = 0.2 s → v(0.2) = 15*0.04 – 6*0.2 + 2 = 0.2 m/s
Practice Problems
- Projectile fired from 20 m height at 25 m/s, 30°. Find total flight time and horizontal distance.
- Particle moves along x-axis: x(t) = t³ – 6t² + 9t. Find times when velocity = 0 and acceleration at those times.
- Ball thrown vertically from cliff 40 m high with 15 m/s. Find max height relative to ground and time to hit base.
- Projectile launched horizontally from 10 m with 12 m/s. Determine impact speed and angle with ground.
- Particle moves along r(t) = (t², t³ – 2t). Find t when particle is at rest and acceleration vector at that instant.