(II) A horse trots away from its trainer in a straight line, moving 38 m away in 9.0 s. It then turns abruptly and gallops halfway back in 1.8 s. calculate (a) its average speed and (b) its average velocity for the entire trip, using “away from the trainer” as the positive direction.
(II) Two locomotives approach each other on parallel tracks. Each has a speed of 55 km/h with respect to the ground. If they are initially 8.5 km apart, how long will it be before they reach each other? (See Fig. 2–35.)
The Correct Answer and Explanation is :
(a) Average Speed:
The average speed is calculated by dividing the total distance traveled by the total time. In this case, the horse moves 38 meters away and then moves halfway back (19 meters). So the total distance traveled is 38 + 19 = 57 meters. The total time taken is 9.0 seconds (away) + 1.8 seconds (back), which equals 10.8 seconds.
Thus, the average speed is:
[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{57 \, \text{m}}{10.8 \, \text{s}} \approx 5.28 \, \text{m/s}
]
(b) Average Velocity:
The average velocity is calculated by dividing the total displacement by the total time. Displacement refers to the difference between the final and initial position. The horse ends up 19 meters away from the trainer (after moving halfway back), so the displacement is 19 meters. The total time is still 10.8 seconds.
Thus, the average velocity is:
[
\text{Average Velocity} = \frac{\text{Displacement}}{\text{Total Time}} = \frac{19 \, \text{m}}{10.8 \, \text{s}} \approx 1.76 \, \text{m/s}
]
(II) Two Locomotives Approaching Each Other:
Each locomotive has a speed of 55 km/h with respect to the ground, and they are initially 8.5 km apart. Since the locomotives are moving towards each other, we need to add their speeds to find the rate at which the distance between them is decreasing.
Total relative speed = 55 km/h + 55 km/h = 110 km/h.
Next, to find how long it will take for the locomotives to reach each other, we divide the distance by the relative speed:
[
\text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{8.5 \, \text{km}}{110 \, \text{km/h}} = 0.07727 \, \text{hours}
]
To convert this to minutes:
[
0.07727 \, \text{hours} \times 60 \, \text{minutes/hour} = 4.64 \, \text{minutes}
]
Thus, it will take approximately 4.64 minutes for the locomotives to reach each other.
Now, I will generate the image for the locomotives’ situation.
Here is the diagram illustrating the locomotives approaching each other on parallel tracks. It shows their relative speed and the time it will take for them to meet. Let me know if you need further clarification!
