prove that average velocity of particle over an interval of time is either smaller than or equal to average speed of particle over same interval?
To prove that the average velocity of a particle over an interval of time is either smaller than or equal to the average speed of the particle over the same interval, we need to use the definitions of velocity and speed and apply some mathematical reasoning.
Let's start by defining velocity and speed:
1. Velocity: Velocity is a vector quantity that represents the rate of change of an object's displacement over time. It has both magnitude and direction. The average velocity is calculated by dividing the change in displacement by the change in time.
2. Speed: Speed is a scalar quantity that represents the rate at which an object covers distance. It only has magnitude. The average speed is calculated by dividing the total distance traveled by the total time taken.
Now, let's consider a particle that undergoes motion over an interval of time. To find the average velocity and average speed, we need to calculate them separately.
Average Velocity:
To find the average velocity, we need the initial and final displacements and the time interval. Let's assume the initial position of the particle is x_i and the final position is x_f. The time interval is denoted by Δt.
The average velocity, V_avg, is given by the formula:
V_avg = (x_f - x_i) / Δt
Average Speed:
To find the average speed, we need the total distance and the time interval. Let's assume the total distance covered by the particle is d.
The average speed, S_avg, is given by the formula:
S_avg = d / Δt
Now, to prove that the average velocity is either smaller or equal to the average speed, we need to compare the two quantities mathematically.
From the formulas above, we can observe that both average velocity and average speed have the same denominator (Δt). This implies that the time interval is the same for both quantities.
To compare the numerators, we need to consider the displacement (x_f - x_i) and the total distance (d). Since displacement is the shortest path between two points, it is always equal to or less than the total distance covered.
Hence, we can conclude that:
(x_f - x_i) ≤ d
Therefore, substituting these inequalities into the average velocity and average speed formulas, we have:
V_avg = (x_f - x_i) / Δt ≤ d / Δt = S_avg
This shows that the average velocity (V_avg) is either smaller than or equal to the average speed (S_avg).
Thus, the average velocity of a particle over an interval of time is either smaller than or equal to the average speed of the particle over the same interval.