A particle moves along the x-axis such that at any time t≥0 its position is x(t), its velocity is v(t), and its acceleration is a(t). You are given:x(0)=6 and v(0)=5Which of the following expressions gives the displacement of the particle over the interval 0≤t≤6 ?∫06v(t)dt∫06∣v(t)∣dt6+∫06∣v(t)∣dt6+∫06v(t)dt
Q. A particle moves along the x-axis such that at any time t≥0 its position is x(t), its velocity is v(t), and its acceleration is a(t). You are given:x(0)=6 and v(0)=5Which of the following expressions gives the displacement of the particle over the interval 0≤t≤6 ?∫06v(t)dt∫06∣v(t)∣dt6+∫06∣v(t)∣dt6+∫06v(t)dt
Define Displacement: Displacement is defined as the change in position of the particle. Since we are given the initial position x(0) and the velocity function v(t), we can find the displacement by integrating the velocity function over the given time interval. The initial position will be the starting point for the displacement.
Find Displacement Interval: The displacement over the interval from 0 to 6 can be found by integrating the velocity function v(t) from 0 to 6. This is because the integral of velocity with respect to time gives the change in position, or displacement.
Calculate Displacement Expression: The correct expression for the displacement is the integral of the velocity function from the start time to the end time, plus the initial position. Therefore, the expression for the displacement is:6+∫06v(t)dt
Consider Absolute Value: The absolute value of the velocity function is not needed because we are looking for the net displacement, not the total distance traveled. The total distance would require the absolute value to ensure all intervals are counted positively, regardless of the direction of motion.
Final Displacement Expression: The expression 6+∫06v(t)dt correctly accounts for the initial position and the change in position due to the velocity over the time interval. This is the expression that gives the displacement of the particle over the interval 0≤t≤6.
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