Let's go step by step to solve part (d). We need to find the total distance the crate travels during the time interval in part (c), where the initial velocity is $-6.60 \, \text{m/s}$ and the final velocity is $13.2 \, \text{m/s}$ with a constant acceleration of $4.70 \, \text{m/s}^2$.

Step 1: Identify when the velocity is zero.

The object first decelerates to zero velocity (stopping momentarily) and then accelerates in the positive direction until it reaches $13.2 \, \text{m/s}$. We can calculate the time it takes for the velocity to reach zero.

Using the equation: $v = v_0 + at$ Set $v = 0$, and solve for $t$: $0 = -6.60 + (4.70)t$ $t = \frac{6.60}{4.70} = 1.404 \, \text{seconds}$

Step 2: Find the displacement from $v_0 = -6.60 \, \text{m/s}$ to $v = 0$.

Now, we find the displacement during this phase (when the object decelerates to rest): $s_1 = v_0 t + \frac{1}{2} a t^2$ $s_1 = (-6.60)(1.404) + \frac{1}{2}(4.70)(1.404)^2$ Calculating: $s_1 = -9.2664 + 4.6297 = -4.6367 \, \text{m}$ So, the displacement until the crate stops is $-4.64 \, \text{m}$.

Step 3: Find the displacement from rest to $v = 13.2 \, \text{m/s}$.

Now, we calculate the displacement after the object starts moving in the positive direction from rest to $13.2 \, \text{m/s}$. Using the same kinematic equation: $v^2 = v_0^2 + 2as$ where $v_0 = 0$, $v = 13.2 \, \text{m/s}$, and $a = 4.70 \, \text{m/s}^2$, we solve for $s_2$: $(13.2)^2 = 0 + 2(4.70)s_2$ $s_2 = \frac{(13.2)^2}{2(4.70)} = \frac{174.24}{9.40} = 18.54 \, \text{m}$

Step 4: Find the total distance.

The total distance is the absolute sum of the distances: $\text{Total distance} = |s_1| + s_2 = 4.64 + 18.54 = 23.18 \, \text{m}$

Final Answer:

The total distance traveled during the interval is $23.18 \, \text{m}$.

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Do you want more details or have any questions?

Here are 5 related questions:

1. How would the displacement change if the acceleration were halved?
2. What is the formula for average velocity during this motion?
3. How would friction impact the calculations in this scenario?
4. How do you calculate the velocity at any given point during the motion?
5. What would happen if the initial velocity and final velocity were both negative?

Tip: Always differentiate between displacement (a vector quantity) and distance (a scalar quantity).