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甲、乙两名同学从同一起点出发，沿平直公路运动到相同终点。甲在前一半时间里以 2m/s做匀速直线运动，后一半时间里以 1m/s 做匀速直线运动；乙在前一半路程...

Mar 19, 2024

甲、乙两名同学从同一起点出发，沿平直公路运动到相同终点。甲在前一半时间里以 2m/s做匀速直线运动，后一半时间里以 1m/s 做匀速直线运动；乙在前一半路程中以1m/s 做匀速直线运动，后一半路程中以2m/s 做匀速直线运动，则甲、乙到达终点的时间之比为（　　） A．1∶1 B．9∶8 C．8∶9 D．4∶3

Solution

Let's denote the total time taken by student 甲 as

T

and the total distance as

D

. For the first half of the time

\frac{T}{2}

, 甲 travels at

2 \text{m/s}

, and for the second half

\frac{T}{2}

, at

1 \text{m/s}

. Calculate the total distance

D

using the average speed formula

The distance covered by 甲 in the first half of the time is

d_1 = 2 \text{m/s} \times \frac{T}{2}

and in the second half is

d_2 = 1 \text{m/s} \times \frac{T}{2}

. The total distance

D

is the sum of

d_1

and

d_2

D = d_1 + d_2

For student 乙, let's denote the time taken to cover the first half of the distance

\frac{D}{2}

1 \text{m/s}

t_1

and the time taken to cover the second half

\frac{D}{2}

2 \text{m/s}

t_2

. Calculate the total time

T'

for 乙 as

T' = t_1 + t_2

The time

t_1

t_1 = \frac{\frac{D}{2}}{1 \text{m/s}}

and

t_2

t_2 = \frac{\frac{D}{2}}{2 \text{m/s}}

. Simplify to find

T'

Compare the total times

T

and

T'

to find the ratio of the times taken by 甲 and 乙 to reach the endpoint

Answer

The ratio of the times taken by 甲 and 乙 to reach the endpoint is 9:8.

Key Concept

Kinematics and the equations of motion for constant velocity are used to solve problems involving time, distance, and velocity.

Explanation

By calculating the distances covered by each student in the given time or for the given part of the journey and then finding the total time taken by each, we can compare their times to find the ratio. The key is to recognize that 甲's velocity changes with time, while 乙's velocity changes with distance.