5 A vehicle is moving in a straight line. Its velocity at different times is recorded and shown below. The velocities are recorded to 5 significant figures and the times may be assumed to be exact.
| Time \(( t\) seconds \()\) | 5 | 10 | 12 | 15 |
| Velocity \(( v\) metres per second \()\) | 5.1250 | 11.000 | 14.000 | 18.375 |
It is suggested initially that a quadratic model may be appropriate for this situation.
- Given that the vehicle is modelled as a particle with constant mass, what assumption about the net force acting on the vehicle leads to a quadratic model?
- Find Newton's interpolating polynomial of degree 2 to model this situation. Write your answer in the form \(v = a t ^ { 2 } + b t + c\).
- Comment on whether this model appears to be appropriate.
- Use this model to find an approximation to the distance travelled over the interval \(5 \leq t \leq 15\).
Further investigation suggests that a cubic model may be more appropriate.
- What technique would you use to fit a cubic model to the data in the table?