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### Analysis of the Problem

The given equation of motion is \( s = at^2 + bt + 10 \), where:
- \( s \) is the height,
- \( t \) is the time,
- \( a \) and \( b \) are constants to be determined.

The table provides the height \( s \) for different values of \( t \), starting with \( s = 45.1 \) at \( t = 0 \). Based on this:

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### Step (a)(i): Determine the values of \( a \) and \( b \)
1. Substitute \( t = 0 \) and \( s = 45.1 \) into the equation:
   \[
   s = at^2 + bt + 10 \implies 45.1 = a(0)^2 + b(0) + 10 \implies 10 = 45.1
   \]

successI’ll begin solving for \(a\) and \(b\). Stay tunedsuccessI've realized I need further user clarification or additional numerical! Shall ProceedTo proceed with determining the constants \(a\) and \(b\) and solving the problem effectively, I need two numerical points from the table. Could you confirm or provide more details from the table, such as another pair of \( (t, s) \) values? This will allow me to set up and solve the required equations for \(a\) and \(b\). 

Let me know if you’d like further explanation! Here are five follow-up questions for better understanding:

1. Would you like me to explain the method of solving for \(a\) and \(b\) using simultaneous equations?
2. Do you need help completing the table once the constants are determined?
3. Would you like a plot of the graph for the relationship between \(s\) and \(t\)?
4. Do you want an explanation of how to interpret the motion of the object from the equation and graph?
5. Are you interested in seeing how calculus (like derivatives) could describe this motion further?

**Tip**: When solving quadratic equations, always try to identify at least two points to set up equations and solve systematically!

The table shows the height in meters of an object thrown vertically upwards varies with the time t seconds. The relationship between s and t is represented by the equation s = at^2 + bt + 10, where a and b are constants. (a)(i) Using the information in the table, determine the values of a and b. (ii) Complete the table. (b)(i) Draw a graph to represent the relationship between s and t.

Analyze a quadratic equation representing the height of an object over time. Solve for constants a and b, complete the table of values, and graph the relationship between height and time. Perfect for students in Grades 9-11.

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