Understanding Velocity Ratios in Constant Acceleration
When a particle starts from rest and moves with constant acceleration, we can analyze the relationship between space-average velocity and time-average velocity.
Definitions
- Space-Average Velocity (V_avg_space): This is defined as the total distance traveled divided by the total time taken. For a particle starting from rest under constant acceleration (a), if it travels a distance (s) in time (t), the formula is:
V_avg_space = s / t
- Time-Average Velocity (V_avg_time): This is the average of the instantaneous velocities during the time interval. Given that the particle starts from rest, the initial velocity (u) is 0. The time-average velocity can be expressed as:
V_avg_time = (u + v) / 2
where v is the final velocity after time t.
Calculating Velocities
- For a particle under constant acceleration:
- The distance traveled (s) is given by the equation:
s = (1/2) * a * t^2
- The final velocity (v) can be expressed as:
v = u + at = at (since u = 0)
Finding the Ratio
1. Space-Average Velocity:
- s = (1/2) * a * t^2
- V_avg_space = s / t = (1/2) * a * t
2. Time-Average Velocity:
- V_avg_time = (0 + at) / 2 = (1/2) * at
Ratio of Velocities
- When we calculate the ratio of space-average velocity to time-average velocity:
Ratio = V_avg_space / V_avg_time = [(1/2) * at] / [(1/2) * at] = 1
Conclusion
The ratio of space-average velocity to time-average velocity for a particle starting from rest with constant acceleration is 1. This indicates that both averages are equal in such scenarios.