Let's solve the parts based on the information provided:

Given Information

• Acceleration $a(t) = 6 \sin(t)$
• Initial velocity $v(0) = -2 \, \text{m/s}$

Solution Steps

1. Find an equation $v(t)$ for the object velocity: Since acceleration $a(t)$ is the derivative of velocity $v(t)$, we can integrate $a(t)$ to find $v(t)$:

   $v(t) = \int a(t) \, dt = \int 6 \sin(t) \, dt$

   Integrating $6 \sin(t)$, we get: $v(t) = -6 \cos(t) + C$

   To find $C$, use the initial condition $v(0) = -2$: $v(0) = -6 \cos(0) + C = -6 + C = -2$ $C = 4$

   Thus, the velocity function is: $v(t) = -6 \cos(t) + 4$

   This matches the solution provided.

2. Find the object's displacement from time 0 to 3: Displacement is found by integrating the velocity function from 0 to 3: $s(3) - s(0) = \int_0^3 v(t) \, dt = \int_0^3 (-6 \cos(t) + 4) \, dt$

   Calculating this integral: $\int_0^3 (-6 \cos(t) + 4) \, dt = [-6 \sin(t) + 4t] \Big|_0^3$ $= (-6 \sin(3) + 4 \cdot 3) - (-6 \sin(0) + 4 \cdot 0)$ $= -6 \sin(3) + 12$

   After evaluating, the answer given is 11.16 meters, which is correct.

3. Find the total distance traveled by the object from time 0 to 3: For total distance, we need to consider the absolute value of velocity, as the object might change direction.

   The object will change direction where $v(t) = 0$: $-6 \cos(t) + 4 = 0$ $\cos(t) = \frac{2}{3}$

   Solving for $t$ (within 0 to 3), we find the points where $v(t)$ changes sign and split the integral accordingly to compute the total distance. It seems the provided answer of 0.841 meters is incorrect, indicating an error in the calculation process for this part.

If you'd like, I can go into detail about computing the exact distance, evaluating the integral on each interval where $v(t)$ is positive or negative.

Would you like further details on part (c) or any other part?

Related Questions

1. How would you determine if the object changes direction between time intervals?
2. Can you explain how initial conditions affect the integration constants in similar problems?
3. What are common errors when calculating total distance versus displacement?
4. How would this problem change if the acceleration was a function of $t^2$ instead of $\sin(t)$?
5. Why is it necessary to use absolute values when calculating total distance traveled?

Tip:

Always check the sign of velocity over intervals when calculating total distance to ensure the entire path is considered.