Section 1.1: Advanced Kinematics
In this section, we explore advanced motion problems including variable acceleration, projectile motion, and relative motion. Vector analysis and graphical methods will also be applied to solve complex kinematics questions.
Key Equations:
- Velocity: \( \vec{v} = \frac{d\vec{r}}{dt} \)
- Acceleration: \( \vec{a} = \frac{d\vec{v}}{dt} \)
- Projectile motion: \( x = v_0 t \cos\theta, \; y = v_0 t \sin\theta - \frac{1}{2} g t^2 \)
- Relative velocity: \( \vec{v}_{AB} = \vec{v}_A - \vec{v}_B \)
Example 1
A particle accelerates along a straight line such that \( a = 2t \) m/s². If initial velocity is 3 m/s, find its velocity at t = 4 s.
\( v = v_0 + \int_0^4 a \, dt = 3 + \int_0^4 2t \, dt = 3 + [t^2]_0^4 = 3 + 16 = 19 \text{ m/s} \)
Example 2
A projectile is launched with speed 20 m/s at 30° above the horizontal. Find its maximum height.
\( H = \frac{(v_0 \sin\theta)^2}{2g} = \frac{(20 \cdot 0.5)^2}{2 \cdot 9.8} = \frac{100}{19.6} \approx 5.10 \text{ m} \)
Practice Problems
- A car accelerates from rest with \( a = 3t \) m/s². Find its velocity after 5 s.
- A projectile is launched at 15 m/s at 45°. Find its range.
- Two cars move along parallel paths. Car A has 20 m/s and Car B has 15 m/s. Find velocity of A relative to B.
- A particle moves in 2D with \( \vec{r} = (4t, 3t^2) \). Find velocity and acceleration at t = 2 s.
- A stone is thrown horizontally from a 25 m high cliff with speed 10 m/s. Find time to hit ground.
- A train decelerates uniformly from 30 m/s to rest in 120 s. Find distance covered.
- A boat moves across a river at 5 m/s while current flows at 3 m/s. Determine resultant velocity.
- A particle moves along x-axis with \( x = 2t^3 - t^2 + 3t \). Find acceleration at t = 2 s.
- A car travels around a circular track of radius 50 m at 20 m/s. Find centripetal acceleration.
- An object falls freely from 50 m. Find its velocity just before hitting the ground.