Understanding the Motion of a Particle on an Inclined Plane
When a particle slides down a frictionless incline, its motion can be described using basic physics principles. The distance travelled by the particle in each second can be analyzed to find the ratio of distances Sn and Sn+1.
Definitions
- Sn: Distance travelled from time = n-1 seconds to time = n seconds.
- Sn+1: Distance travelled from time = n seconds to time = n+1 seconds.
Key Concepts
- The particle accelerates down the incline due to gravity. The acceleration (a) can be defined as a = g sin(θ), where g is the acceleration due to gravity and θ is the angle of incline.
- The distances covered in each time interval can be calculated using the equation of motion:
- Distance = Initial velocity × Time + 0.5 × Acceleration × Time^2.
Distance Calculation
- For Sn (from t=n-1 to t=n):
- Initial velocity = 0 (since it starts sliding).
- Time = 1 second.
- Distance Sn = 0.5 × a × (1^2) = 0.5a.
- For Sn+1 (from t=n to t=n+1):
- Initial velocity = a (velocity at the end of the first second).
- Time = 1 second.
- Distance Sn+1 = a + 0.5 × a × (1^2) = a + 0.5a = 1.5a.
Finding the Ratio
- The ratio Sn/Sn+1 can thus be calculated:
Ratio Sn/Sn+1 = (0.5a) / (1.5a) = 0.5 / 1.5 = 1/3.
Conclusion
- The ratio of distances travelled in successive seconds as the particle slides down the incline is 1:3. This highlights how the distance covered increases as time progresses due to constant acceleration.