The table shows the position of the cyclist
t (seconds) | 0 | 1 | 2 | 3 | 4 | 5 |
s (meters) | 0 | 1.1 | 4.6 | 10.1 | 17.9 | 25.5 |
(a) Find the average velocity for each time period.
(i) [1, 3] __ m/s
(ii) [2, 3] __ m/s
(iii) [3, 5] __ m/s
(iv) [3, 4] __ m/s
(b) Estimate the instantaneous velocity when t = 3.
__ m/s
(a) To find the average velocity for each time period, you can use the formula for average velocity:
Average Velocity (m/s) = Δs / Δt
Where Δs is the change in position (final position - initial position) and Δt is the change in time (final time - initial time).
(i) [1, 3]: Δs = s(3) - s(1) = 10.1 - 1.1 = 9.0 meters Δt = 3 - 1 = 2 seconds
Average Velocity = Δs / Δt = 9.0 m / 2 s = 4.5 m/s
(ii) [2, 3]: Δs = s(3) - s(2) = 10.1 - 4.6 = 5.5 meters Δt = 3 - 2 = 1 second
Average Velocity = Δs / Δt = 5.5 m / 1 s = 5.5 m/s
(iii) [3, 5]: Δs = s(5) - s(3) = 25.5 - 10.1 = 15.4 meters Δt = 5 - 3 = 2 seconds
Average Velocity = Δs / Δt = 15.4 m / 2 s = 7.7 m/s
(iv) [3, 4]: Δs = s(4) - s(3) = 17.9 - 10.1 = 7.8 meters Δt = 4 - 3 = 1 second
Average Velocity = Δs / Δt = 7.8 m / 1 s = 7.8 m/s
(b) To estimate the instantaneous velocity at t = 3 seconds, you can look at the average velocity over a very short time interval that includes t = 3. In this case, you can use the interval [2, 3] because it's very close to t = 3.
Δs = s(3) - s(2) = 10.1 - 4.6 = 5.5 meters Δt = 3 - 2 = 1 second
Instantaneous Velocity ≈ Δs / Δt ≈ 5.5 m / 1 s = 5.5 m/s
So, the estimated instantaneous velocity at t = 3 seconds is approximately 5.5 m/s.
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