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(2)=4 and v(2)=6Which of the following expression gives the velocity of the particle when t=9?6+∫92a(t)dt6+∫92v(t)dt6+∫29a(t)dt6+∫29v(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(2)=4 and v(2)=6Which of the following expression gives the velocity of the particle when t=9?6+∫92a(t)dt6+∫92v(t)dt6+∫29a(t)dt6+∫29v(t)dt
Understand Problem: To find the velocity of the particle at t=9, we need to use the information given about the particle's velocity at t=2 and how the velocity changes over time, which is given by the acceleration a(t).
Calculate Initial Velocity: We know the velocity at t=2 is v(2)=6. To find the velocity at t=9, we need to add the change in velocity from t=2 to t=9 to the initial velocity at t=2. The change in velocity is given by the integral of the acceleration function a(t) from t=2 to t=9.
Determine Change in Velocity: The correct expression for the velocity at t=9 is the initial velocity plus the integral of the acceleration from t=2 to t=9. This is represented by the expression:v(9)=v(2)+∫t=2t=9a(t)dt
Calculate Final Velocity: Therefore, the correct expression from the given options is: 6+∫t=2t=9a(t)dt
Verify Correctness: The other options are incorrect because they either have the wrong limits of integration or are integrating the wrong function (velocity instead of acceleration).
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