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Identify the velocity function and check for changes in direction.

Computer Science

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Answer

3 m\text{3 m}

Step-by-step solution

3 steps
  1. Identify the velocity function and check for changes in direction.

    The velocity of the particle is given by v=(−2t+4) m/sv = (-2t + 4) \text{ m/s}.
    To determine if the particle changes direction within the first second (t=0t=0 to t=1t=1), we find when v=0v = 0.

    −2t+4=0-2t + 4 = 0 2t=42t = 4 t=2 st = 2 \text{ s}

    Since t=2 st = 2 \text{ s} is outside the interval [0,1][0, 1] s, the particle does not change direction during the first second. Therefore, the distance moved is simply the absolute displacement.

  2. Find the position function.

    Position ss is the integral of velocity with respect to time tt.

    s=∫v dt=∫(−2t+4) dts = \int v \, dt = \int (-2t + 4) \, dt s=−t2+4t+Cs = -t^2 + 4t + C

    For calculating the distance moved, we can assume the initial position at t=0t=0 is s=0s=0.

    0=−(0)2+4(0)+C  ⟹  C=00 = -(0)^2 + 4(0) + C \implies C = 0

    So, the position function is:

    s(t)=−t2+4ts(t) = -t^2 + 4t
  3. Calculate the distance moved during the first second.

    The distance moved is the absolute difference between the position at t=1t=1 and the position at t=0t=0.

    s(1)=−(1)2+4(1)=−1+4=3 ms(1) = -(1)^2 + 4(1) = -1 + 4 = 3 \text{ m} s(0)=−(0)2+4(0)=0 ms(0) = -(0)^2 + 4(0) = 0 \text{ m}

    Distance moved =∣s(1)−s(0)∣=∣3−0∣=3 m= |s(1) - s(0)| = |3 - 0| = 3 \text{ m}.
    The distance moved by the particle during the first second of its motion is 3 m\boxed{\text{3 m}}.

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