To solve this problem, let's go step by step.

The given data is:

• Acceleration $a = -1.77 \, \text{m/s}^2$
• Time $t = 3.33 \, \text{s}$
• Final velocity $v_f = -8.77 \, \text{m/s}$

We need to find the displacement $\Delta x$. The best approach would be to use the equation that relates displacement, initial velocity, acceleration, and time.

From the data, we do not have the initial velocity ($v_i$) directly, but we know the final velocity, time, and acceleration. The equation that works for this scenario is the one that combines these variables. Checking the options:

• Option A: $\Delta x = v_f t - \frac{1}{2} a t^2$ This could be correct since we know $v_f$, $a$, and $t$.

• Option B: $\Delta x = \frac{1}{2}(v_f + v_i) t$ This requires $v_i$, which we don't have directly.

• Option C: $v_f = v_i + at$ This gives the final velocity in terms of initial velocity but doesn't solve for displacement.

• Option D: $v_f^2 = v_i^2 + 2a\Delta x$ This can solve for displacement but requires both initial velocity and final velocity.

• Option E: $\Delta x = v_i t + \frac{1}{2} a t^2$ This is a standard displacement equation, but we don't have $v_i$.

Conclusion: Since we have final velocity, acceleration, and time, the best equation is Option A, as it allows us to solve for displacement directly with the known values.

Would you like me to go into the details of solving the displacement using this formula, or do you have any other questions?

Here are five relative questions that can help deepen your understanding:

1. How can you calculate the initial velocity if you only know the final velocity, time, and acceleration?
2. Why is the negative acceleration important in this problem?
3. What happens to displacement if the acceleration is positive instead of negative?
4. How do you interpret the significance of the time factor in uniform acceleration problems?
5. What real-life situations could mirror this kind of motion?

Tip: When working with equations of motion, it's crucial to pick the equation that incorporates all the known quantities, so you can solve for the unknown directly.