Final answer:
Without the function describing the curve, we cannot directly calculate the unit tangent vector or the curve's length. The slope of the tangent can be found using endpoint positions and times. The acceleration can be calculated using endpoint velocities and times.
Step-by-step explanation:
Finding the Tangent Vector:
To find the unit tangent vector of a curve, we need the function that describes the curve and a specific point in time. However, as the provided information does not include the actual functional form of the curve, we can only provide a general approach for such a calculation.
Normally, you would take the derivative of the curve's function with respect to time, evaluate it at t = 25 s, and then normalize this vector to obtain the unit tangent vector.
Calculating the Length of the Curve:
To calculate the length of the indicated portion of the curve, which is the distance between two points on the curve, we usually integrate the speed function over the given time interval. Again, without the specific function, we can't perform this calculation.
Understanding Slope and Displacement:
When given two endpoints, such as a position of 1300 m at time 19 s and a position of 3120 m at time 32 s, you can calculate the slope of the tangent by taking the difference in the positions divided by the difference in times to find the average velocity over that interval.
Determining Vector Resultant:
The slope is also associated with the acceleration if given the respective velocities at two different points in time, such as (260 m/s - 210 m/s)/(51 s - 1.0 s), resulting in an acceleration of 1.0 m/s².
The complete question is: How To Find The Unit Tangent Vector?