the graph for the first 5 seconds is 1m/1s then forth is 5m/1s untill 10 seconds where the graph ends, write the equation for the position (x) in terms of time (t) for the time interval t = 5 to 10 seconds

To find the equation for the position \( x \) in terms of time \( t \) for the time interval \( t = 5 \) to \( t = 10 \) seconds, given the graph's behavior, we can break down the problem as follows:

### Problem Breakdown

1. **From \( t = 0 \) to \( t = 5 \) seconds:**
   - The object moves at a constant speed of \( 1 \, \text{m/s} \).
   - The position function for this interval is:
     \[
     x(t) = 1t \quad \text{(since speed is constant, speed = slope = 1 m/s)}
     \]
   - At \( t = 5 \) seconds, the position is:
     \[
     x(5) = 1 \times 5 = 5 \, \text{m}
     \]

2. **From \( t = 5 \) to \( t = 10 \) seconds:**
   - The object moves at a constant speed of \( 5 \, \text{m/s} \).
   - We need to find the position function for this interval, taking into account that the motion is continuous and starts from \( x = 5 \, \text{m} \) at \( t = 5 \, \text{s} \).

### Finding the Position Function for \( t = 5 \) to \( t = 10 \) Seconds

Since the object continues moving at a speed of \( 5 \, \text{m/s} \) from \( t = 5 \) seconds:

1. **Initial position at \( t = 5 \):** \( x = 5 \) meters.
2. **Speed from \( t = 5 \) to \( t = 10 \):** \( 5 \, \text{m/s} \).

The position function can be expressed as:

\[
x(t) = x(5) + \text{(speed)} \times \text{(time elapsed since } t = 5 \text{)}
\]

For \( t = 5 \) to \( t = 10 \):

\[
x(t) = 5 + 5 \times (t - 5)
\]

Here’s the reasoning:
- At \( t = 5 \), the position is \( x = 5 \, \text{m} \).
- For every additional second \( t - 5 \), the object moves \( 5 \) meters.

### Final Position Equation

\[
x(t) = 5 + 5(t - 5)
\]

Simplify the equation:

\[
x(t) = 5 + 5t - 25
\]

\[
x(t) = 5t - 20
\]

This is the equation for the position \( x \) in terms of time \( t \) for the time interval \( t = 5 \) to \( t = 10 \) seconds.

Would you like further details or have any questions?

### Related Questions

1. What would the graph of \( x(t) \) look like for the entire time interval \( t = 0 \) to \( t = 10 \) seconds?
2. How would the position function change if the speed after 5 seconds was \( 3 \, \text{m/s} \) instead of \( 5 \, \text{m/s} \)?
3. What is the average speed of the object over the entire 10 seconds?
4. How far does the object travel between \( t = 5 \) and \( t = 10 \) seconds?
5. Can you find the acceleration of the object if the speed changes instantaneously at \( t = 5 \) seconds?

### Tip

When dealing with piecewise functions, make sure to match conditions at the boundaries (e.g., \( t = 5 \) in this problem) to ensure continuity and correctly express each part of the function over the relevant intervals.

Explore the position equation for time interval 5 to 10 seconds with a detailed step-by-step solution. This problem involves kinematics and constant speed concepts, providing insights into how to derive and interpret position functions over specified time intervals. Suitable for high school students studying physics or advanced mathematics.

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