Question

Движение двух автомобилей по шоссе заданы уравнениями xцыфра снизу(1)=3+1,5t^2 и xцыфра снизу(2)=7-4t.Найдите время и место встречи автомобилей и расстояние между ними через 5 с от начала отсчета времени.

Answer 1

1)
при встрече x₁=x₂
значит
3+1.5t²=7-4t
1.5t²+4t-4=0
корни t₁=-3.44   t₂=0.77
Поскольку время не может быть отрицательным выбираем t₂
Встретятся через 0.77 сек в координате 7-4t=7-4·0.77=3.92
 
2)
подставляем 5сек в каждое уравнение и получаем
х₁= 40.5
х₂= -13
S=x₁- x₂=53.5

Answer 2

Problem Analysis

We are given the equations of motion for two cars on a highway: x(1) = 3 + 1.5t^2 and x(2) = 7 - 4t. We need to find the time and location of the cars' meeting point, as well as the distance between them after 5 seconds.

Finding the Time of Meeting

To find the time of meeting, we need to set the two equations equal to each other and solve for t:

3 + 1.5t^2 = 7 - 4t

Let's solve this equation:

3 + 1.5t^2 + 4t - 7 = 0

1.5t^2 + 4t - 4 = 0

We can solve this quadratic equation using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1.5, b = 4, and c = -4. Plugging in these values, we get:

t = (-4 ± √(4^2 - 4 * 1.5 * -4)) / (2 * 1.5)

Simplifying further:

t = (-4 ± √(16 + 24)) / 3

t = (-4 ± √40) / 3

t = (-4 ± 2√10) / 3

Therefore, the two possible values for t are:

t1 = (-4 + 2√10) / 3 t2 = (-4 - 2√10) / 3

We need to determine which value of t is valid based on the context of the problem. Since time cannot be negative, we can discard t2. Therefore, the time of meeting is t1.

Finding the Location of Meeting

To find the location of the meeting, we substitute the value of t1 into either of the equations. Let's use x(1):

x(1) = 3 + 1.5t^2 x(1) = 3 + 1.5 * ((-4 + 2√10) / 3)^2

Simplifying further:

x(1) = 3 + 1.5 * ((-4 + 2√10)^2 / 9)

x(1) = 3 + 1.5 * (16 - 16√10 + 40 / 9)

x(1) = 3 + 1.5 * (56 - 16√10) / 9

x(1) = 3 + (84 - 24√10) / 9

x(1) = (27 + 84 - 24√10) / 9

x(1) = (111 - 24√10) / 9

Therefore, the location of the meeting is (111 - 24√10) / 9.

Finding the Distance between the Cars after 5 seconds

To find the distance between the cars after 5 seconds, we need to calculate the positions of each car at that time and then find the difference.

Let's calculate the positions of each car at t = 5:

x(1) = 3 + 1.5 * 5^2 x(1) = 3 + 1.5 * 25 x(1) = 3 + 37.5 x(1) = 40.5

x(2) = 7 - 4 * 5 x(2) = 7 - 20 x(2) = -13

Therefore, the positions of the cars at t = 5 are x(1) = 40.5 and x(2) = -13. The distance between them is:

Distance = |x(1) - x(2)| Distance = |40.5 - (-13)| Distance = |40.5 + 13| Distance = 53.5

Therefore, the distance between the cars after 5 seconds is 53.5 units.

Summary

Based on the given equations of motion, we have determined the following:

- The time of meeting between the two cars is t = (-4 + 2√10) / 3. - The location of the meeting is (111 - 24√10) / 9. - The distance between the cars after 5 seconds is 53.5 units.