A particle moves along the x-axis. The velocity of the particle at time t is 6t – t^2. What is the total distance traveled by the particle from time t=0 to t=3?
To find the total distance traveled by the particle, we need to calculate the area under the velocity-time graph from time t=0 to t=3
To find the total distance traveled by the particle, we need to calculate the area under the velocity-time graph from time t=0 to t=3. Since the velocity function is given as 6t – t^2, we first need to find the integral of this function over the interval [0, 3].
Let’s find the antiderivative of the velocity function:
∫(6t – t^2) dt = 3t^2 – (t^3/3) + C,
where C is the constant of integration.
Now, we can evaluate the definite integral from t=0 to t=3:
∫[(6t – t^2) dt] from 0 to 3 = [3t^2 – (t^3/3)] evaluated from 0 to 3
= [(3*3^2 – (3^3/3))] – [(3*0^2 – (0^3/3))]
= [27 – 9] – [0 – 0]
= 18.
Therefore, the total distance traveled by the particle from time t=0 to t=3 is 18 units.
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