Answer:
To find the displacement of the particle, we need to calculate the definite integral of the velocity function over the given time interval.
(a) Displacement:
The displacement of the particle is given by the definite integral of the velocity function from t = 1 to t = 6:
Displacement = ∫[1 to 6] v(t) dt
First, let's find the antiderivative of the velocity function:
∫(-2(t^2 - 2t - 8)) dt = -2∫(t^2 - 2t - 8) dt
Integrating each term separately:
= -2 * [ (1/3)t^3 - t^2 - 8t ] evaluated from t = 1 to t = 6
= -2 * [ (1/3)(6^3) - (6^2) - 8(6) - (1/3)(1^3) + (1^2) + 8(1) ]
= -2 * [ (216/3) - 36 - 48 - (1/3) + 1 + 8 ]
= -2 * [ 72 - 36 - 48 - 1 + 1 + 8 ]
= -2 * [ -4 ]
= 8 meters
Therefore, the displacement of the particle is 8 meters.
(b) Total distance traveled:
To find the total distance traveled by the particle, we need to consider the absolute value of the velocity function over the given time interval.
Total distance traveled = ∫[1 to 6] |v(t)| dt
First, let's find the absolute value of the velocity function:
|v(t)| = |-2(t^2 - 2t - 8)| = 2|t^2 - 2t - 8|
Now, let's find the integral of the absolute value of the velocity function:
∫[1 to 6] 2|t^2 - 2t - 8| dt
To evaluate this integral, we need to split the integral into two parts based on the intervals where the function inside the absolute value changes sign.
For t = 1 to t = 3:
∫[1 to 3] 2(t^2 - 2t - 8) dt = 2∫[1 to 3] (t^2 - 2t - 8) dt
Integrating each term separately:
= 2 * [ (1/3)t^3 - t^2 - 8t ] evaluated from t = 1 to t = 3
= 2 * [ (1/3)(3^3) - (3^2) - 8(3) - (1/3)(1^3) + (1^2) + 8(1) ]
= 2 * [ (27/3) - 9 - 24 - (1/3) + 1 + 8 ]
= 2 * [ 9 - 9 - 24 - 1 + 1 + 8 ]
= 2 * [ -16 ]
= -32 meters (taking into account the negative sign)
For t = 3 to t = 6:
∫[3 to 6] 2(t^2 - 2t - 8) dt = 2∫[3 to 6] (t^2 - 2t - 8) dt
Integrating each term separately:
= 2 * [ (1/3)t^3 - t^2 - 8t ] evaluated from t = 3 to t = 6
= 2 * [ (1/3)(6^3) - (6^2) - 8(6) - (1/3)(3^3) + (3^2) + 8(3) ]
= 2 * [ (216/3) - 36 - 48 - (27/3) + 9 + 24 ]
= 2 * [ 72 - 36 - 48 - 9 + 9 + 24 ]
= 2 * [ 12 ]
= 24 meters
Adding the distances together:
Total distance traveled = |-32| + 24 = 32 + 24 = 56 meters
Therefore, the total distance traveled by the particle is 56 meters.