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To solve this problem, we need to determine the total acceleration \(\alpha\) of the minivan after 5 seconds. The acceleration in this scenario has two components:

1. **Tangential acceleration (\(a_t\))** due to the constant forward acceleration.
2. **Centripetal (or normal) acceleration (\(a_n\))**, which depends on the velocity and the radius of curvature of the path.

### 1. Tangential acceleration (\(a_t\)):
This is given by the problem as \(a_t = 1.8 \, \text{m/s}^2\).

The velocity after 5 seconds due to the tangential acceleration is:
\[
v = a_t \cdot t = 1.8 \, \text{m/s}^2 \times 5 \, \text{s} = 9 \, \text{m/s}
\]

### 2. Normal acceleration (\(a_n\)):
The normal (centripetal) acceleration is given by:
\[
a_n = \frac{v^2}{\rho}
\]
where \(v = 9 \, \text{m/s}\) and \(\rho = 40 \, \text{m}\) is the radius of curvature.

Substituting these values:
\[
a_n = \frac{(9 \, \text{m/s})^2}{40 \, \text{m}} = \frac{81}{40} = 2.025 \, \text{m/s}^2
\]

### 3. Total acceleration (\(\alpha\)):
The total acceleration is the vector sum of the tangential and normal accelerations. Since these accelerations are perpendicular to each other, we can calculate the magnitude of the total acceleration using the Pythagorean theorem:
\[
\alpha = \sqrt{a_t^2 + a_n^2}
\]
Substituting the values:
\[
\alpha = \sqrt{(1.8)^2 + (2.025)^2} = \sqrt{3.24 + 4.100625} = \sqrt{7.340625} \approx 2.71 \, \text{m/s}^2
\]

Thus, the magnitude of the total acceleration after 5 seconds is approximately **2.71 m/s²**.

Would you like further details or explanations?

Here are 5 related questions:
1. How does the banking angle affect the centripetal acceleration?
2. How would the result change if the radius of curvature were larger?
3. What is the effect of the forward acceleration on the velocity after 10 seconds?
4. How would the total acceleration change if the vehicle had an increasing forward acceleration instead of a constant one?
5. What role does friction play in real-world applications of banking curves?

**Tip:** In problems with both tangential and centripetal acceleration, always resolve the components separately before combining them.

A minivan starts from rest on a road whose constant radius of curvature is 40 m and whose bank angle is 10°. The motion occurs in a horizontal plane. If the constant forward acceleration of the minivan is 1.8 m/s², determine the magnitude α of its total acceleration 5 seconds after starting.

Learn how to determine the total acceleration of a minivan moving on a banked road with a radius of curvature of 40 m and a forward acceleration of 1.8 m/s². This physics problem involves calculating tangential and centripetal accelerations and using vector addition for total acceleration.

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