### Shortcut Methods

**Typical Numerical for JEE Exams:**

**1. Distance and Displacement:**

**Shortcut:**

- Distance = Final position - Initial position
- Displacement = Final position - Initial position (in a specified direction)

**Solution:**

- Distance = 5m - (-2m) = 7m
- Displacement = 5m - (-2m) = 7m (in the positive direction)

**2. Average Speed and Average Velocity:**

**Shortcut:**

- Average speed = Total distance traveled / Total time taken
- Average velocity = Displacement / Total time taken

**Solution:**

- Average speed = 200 km / 4 hr = 50 km/hr
- Since the car returns to its starting point, the displacement is 0 km.
- Average velocity = 0 km / 4 hr = 0 km/hr

**3. Instantaneous Speed and Instantaneous Velocity:**

**Shortcut:**

- Instantaneous speed = ds/dt (derivative of position with respect to time)
- Instantaneous velocity = dv/dt (derivative of displacement with respect to time)

**Solution:**

- v(2) = ds/dt = 6(2)^2-6(2)+4 = 16 m/s
- v(2) = dv/dt = 12t - 6 = 18 m/s

**4. Acceleration:**

**Shortcut:**

- Acceleration = (Final velocity - Initial velocity) / Time taken
- v = u + at

**Solution:**

- a = (0 - 0) / 10 = 0 m/s^2
- v = 0 + 5 * 10 = 50 m/s
- Displacement = (u + v)/2 * t = (0 + 50) / 2 * 10 = 250m

**5. Motion with Constant Acceleration:**

**Shortcut:**

- v = u + at
- s = ut + 1/2 at^2

**Solution:**

- v = 0 + (-5 * 36/5 * 18/5) = -18 m/s
- s = 0 + 1/2 * (-5 * 36/5 * 18/5)^2 = -9/2 km = -4.5 km

**6. Relative Motion:**

**Shortcut:**

- Velocity of passenger relative to ground = Velocity of car + Velocity of passenger relative to car
- Relative velocity = Vcar + Vpassenger (or Vcar - Vpassenger)

**Solution:**

- Velocity of passenger relative to car: 2 m/s (towards west)
- Velocity of passenger relative to ground: 60 km/hr north + 2 m/s west

**Typical Numerical for CBSE Board Exams:**

**1. Distance and Displacement:**

- Distance = 2 km + 3 km = 5 km
- Displacement = √(2^2 + 3^2) km = √13 km

**2. Average Speed:**

- Average speed = 120 km / 2 hr = 60 km/hr

**3. Acceleration:**

- v = u + at
- v = 0 + 10 * 5 = 50 m/s

**4. Motion with Constant Acceleration:**

- v = 72 * 5/18 - 10 = 20 m/s
- s = 72 * 5/18 * 10 - 1/2 * 10^2 = 150 m

**5. Relative Motion:**

- Velocity of fish relative to boat: 15 km/hr west - 5 km/hr south
- Velocity of fish relative to water: √(15^2 + 5^2) km/hr = 20 km/hr at an angle of 18.43
^{o}south of west