University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

488

Pages:

5
Author:

Jabin F.

Speed (v) = rate of change of distance with respect to time. (instantaneous) v = ds/dt average speed = total distance / total time = S/t ⇒ If speed is uniform then v = = S/t Velocity (v) = rate of change of displacement or position vector. (instantaneous) v = dr/dt = dx/dt = dy/dt ⇒ magnitude of instantaneous velocity = instantaneous speed |v| = |dr/dt| = v Average velocity = total displacement / total time = 1/t ∫v dt ⇒ || ≤ <|v|>; only for straight line motion in 1-dimensional. Example: Instantaneous velocity of a particle moving on an x-y plane is v = cos(t) i + sin(t) j. What is the distance moved from time t=0 to t=π? Solution: r = ∫v dt = ∫(cos(t) i + sin(t) j) dt = sin(t) i - cos(t) j |_0^π = 2i |dr| = |v| dt S = ∫|dr| = ∫|v| dt = ∫√(cos²(t) + sin²(t)) dt = ∫dt = t |_0^π = π Q: If v = t i + t² j. Then calculate distance moved from time t=0 to t=1 sec. Solution: |v| = √(t² + t⁴) ds = |v| dt or ds = √(t² + t⁴) dt S = ∫√(t² + t⁴) dt = ∫t√(1 + t²) dt Let t² = x ⇒ 2t dt = dx ⇒ t dt = dx/2 When t = 0 ⇒ x = 0 When t = 1 ⇒ x = 1 ∴ S = ∫(x + x³/2) dx = ½ [x² + (x⁴/4)] |_0¹ = ½ (1 + ¼) = 5/8 Ans.