A horse canters away from its trainer in a straight line, moving 140 m away in 16.0 s. It then turns abruptly and gallops halfway back in 5.0 s.

(a) Calculate its average speed.
(b) Calculate its average velocity for the entire trip, using "away from the trainer" as the positive direction.

8.88 meters per second

To solve this problem, we will first calculate the average speed and then the average velocity of the horse.

(a) Average speed:
The average speed of an object is given by the formula:

Average Speed = Total Distance / Total Time

In this case, the horse moves away from its trainer in a straight line and then turns back halfway. So, we need to calculate the total distance covered and the total time taken for the entire trip.

Total Distance = Distance away from trainer + Distance towards trainer
= 140 m + (1/2) * 140 m (since the horse turns back halfway)
= 140 m + 70 m
= 210 m

Total Time = Time taken to move away + Time taken to move towards
= 16.0 s + 5.0 s
= 21.0 s

Now, we can use the formula for average speed:

Average Speed = Total Distance / Total Time
= 210 m / 21.0 s
= 10.0 m/s

Therefore, the average speed of the horse for the entire trip is 10.0 m/s.

(b) Average velocity:
Velocity is a vector quantity that takes into account the direction of motion. Since the problem specifies "away from the trainer" as the positive direction, we'll use it to calculate the average velocity.

Average Velocity = Total Displacement / Total Time

Displacement is the difference between the initial position and final position. In this case, the horse moved away from the trainer in a straight line and then came back halfway, ending up 70 m away from the trainer.

Total Displacement = Final Position - Initial Position
= 70 m - 140 m
= -70 m

Note: A negative displacement indicates motion in the opposite direction to the chosen positive direction.

Now, we can use the formula for average velocity:

Average Velocity = Total Displacement / Total Time
= -70 m / 21.0 s
= -3.33 m/s (rounded to two decimal places)

Therefore, the average velocity of the horse for the entire trip, considering "away from the trainer" as the positive direction, is -3.33 m/s.