The following equation describes an object's motion.  The object moves along a line with uniform acceleration.  What is the starting position of the object?v(t)=2.6t-5

The equation provided describes the velocity \( v(t) \) of an object as a function of time \( t \):

\[
v(t) = 2.6t - 5
\]

To find the starting position of the object, we need to determine the position function \( s(t) \). We know that velocity is the derivative of position with respect to time:

\[
v(t) = \frac{ds(t)}{dt}
\]

To find the position function \( s(t) \), we integrate the velocity function:

\[
s(t) = \int v(t) \, dt = \int (2.6t - 5) \, dt
\]

Integrating the expression:

\[
s(t) = \int 2.6t \, dt - \int 5 \, dt
\]

\[
s(t) = 2.6 \cdot \frac{t^2}{2} - 5t + C
\]

\[
s(t) = 1.3t^2 - 5t + C
\]

Here, \( C \) is the constant of integration, which represents the initial position of the object.

To find the starting position (i.e., when \( t = 0 \)), we substitute \( t = 0 \) into the position function:

\[
s(0) = 1.3(0)^2 - 5(0) + C = C
\]

So, the starting position of the object is \( C \).

Without additional information, \( C \) remains an unknown constant, but it represents the object's position at \( t = 0 \).

Would you like more details or have any questions?

Here are some related questions:
1. How do you determine the acceleration of the object from the velocity function?
2. What would the position function look like if the initial velocity was different?
3. How does the initial velocity affect the object's motion over time?
4. What role does the constant \( C \) play in determining the object's trajectory?
5. How would the velocity function change if the object had non-uniform acceleration?

**Tip:** The constant of integration \( C \) is crucial for determining initial conditions, such as starting position or initial velocity.

Explore how to find the starting position of an object with uniform acceleration described by the velocity function v(t)=2.6t-5. Learn about calculus concepts like integration and their application in kinematics. Suitable for advanced high school students and learners interested in physics.

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