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PS1-5 - Problem Set 1 Problem 5 Problem The acceleration of...

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Problem Set 1: Problem 5. Problem: The acceleration of a particle is a = kt , where k is a constant to be determined. Its position and velocity at t = 0 are x (0) = 0 and v (0) = v o , respectively, where v o > 0 . The particle’s velocity at time t = τ is v ( τ ) = 8 9 v o . (a) Determine the constant k as a function of v o and τ . Based on your solution for k , express the velocity as a function of t , τ and v o . (b) Solve for the particle’s position as a function of t , τ and v o . (c) What is the total distance traveled by the particle from t = 0 to t = 9 τ ? NOTE: Be sure to account for the fact that the particle changes direction during this time interval. Solution: Since we know the velocity at two different times, the obvious first step in solving this problem is to integrate the acceleration over time to determine the velocity. That will yield sufficient information to determine k . (a) The velocity is given by v ( t ) = 8 t 0 kt dt = v o + 1 2 kt 2 Now, we make use of the fact that v ( τ ) = 8 9 v o , i.e., v o + 1 2 k τ 2 = 8 9 v o Solving for k yields k =
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