Question

Лодка замедляет свое движение под действием сопротивления воды, которое пропорционально скорости лодки. Начальная скорость лодки 1,5 м/сек, через 4 сек скорость ее 1 м/сек. Когда скорость уменьшится до 1 см/сек? Какой путь может пройти лодка до остановки. решение расписать​

Answer 1

Ответ:

14.8м

Объяснение:

Пусть v(t) - скорость лодки в момент времени t. Тогда, согласно второму закону Ньютона:

mv′=F

где m - масса лодки, F - сила сопротивления воды.

Так как скорость лодки пропорциональна сопротивлению воды, то F=kv. Получаем уравнение:

v′=kmv=k1v.

Это уравнение является уравнением с разделяющимися переменными. Разделяя переменные и интегрируя получим общее решение:

v=Cek1t

Подставив начальные условия v(0)=1,5 и v(4)=1, получим:

C=1,5; k1=14ln(23).

Получаем, скорость движения лодки:

v=1,5e0,25ln(2/3)t

Подставляя v=1см/сек=0,01м/сек, получим время:

0,01=1,5e0,25ln(2/3)t

0,25ln(2/3)t=ln(0,01/1,5)

t=4ln(0,01/1,5)ln(2/3)≈49.43

Таким образом, скорость уменьшится до 1 см/сек через 49.43 сек.

Так как скорость, это производная от пути, то есть s′(t)=v(t). Найдем s(t):

s=∫1,5e0,25ln(2/3)t=1,50,25ln(2/3)e0,25ln(2/3)t+C

Так как s(0)=0, то получаем:

C=−1,50,25ln(2/3) ⇒ s=1,50,25ln(2/3)(e0,25ln(2/3)t−1)

Так как v(t)=0 при t→∞, то путь который лодка пройдет до остановки равен:

limt→∞s(t)=−1,50,25ln(2/3)≈14.8

Таким образом, путь который лодка пройдет до остановки: 14.8 м.

Answer 2

Problem Statement

A boat slows down due to the resistance of the water, which is proportional to the boat's speed. The initial speed of the boat is 1.5 m/s, and after 4 seconds, its speed is 1 m/s. We need to determine when the speed of the boat will decrease to 1 cm/s and calculate the distance the boat can travel before coming to a stop.

Solution

To solve this problem, we can use the concept of exponential decay. The equation that describes the relationship between the speed of the boat and time is:

v(t) = v0 * e^(-kt)

Where: - v(t) is the speed of the boat at time t - v0 is the initial speed of the boat - k is a constant that depends on the resistance of the water

We are given that the initial speed of the boat is 1.5 m/s and after 4 seconds, its speed is 1 m/s. We can use these values to find the value of k.

Using the equation v(t) = v0 * e^(-kt), we can substitute the given values:

1 = 1.5 * e^(-4k)

Simplifying the equation, we get:

e^(-4k) = 1/1.5

Taking the natural logarithm of both sides, we get:

-4k = ln(1/1.5)

Solving for k, we have:

k = -ln(1/1.5)/4

Now that we have the value of k, we can use it to find when the speed of the boat will decrease to 1 cm/s. Let's denote this time as t_stop.

1 cm/s = 1.5 * e^(-kt_stop)

Simplifying the equation, we get:

e^(-kt_stop) = 1/1.5

Taking the natural logarithm of both sides, we get:

-kt_stop = ln(1/1.5)

Solving for t_stop, we have:

t_stop = -ln(1/1.5)/(k)

Now that we have the value of t_stop, we can calculate the distance the boat can travel before coming to a stop. The distance traveled is given by the equation:

d = ∫[0, t_stop] v(t) dt

Substituting the equation for v(t), we get:

d = ∫[0, t_stop] (v0 * e^(-kt)) dt

Integrating the equation, we get:

d = v0/k * (1 - e^(-kt_stop))

Now we can substitute the values of v0, k, and t_stop to calculate the distance traveled by the boat before coming to a stop.

Calculation

Given: - v0 = 1.5 m/s - t_stop = -ln(1/1.5)/(k)

We need to calculate: - t_stop (time when the speed decreases to 1 cm/s) - d (distance traveled by the boat before coming to a stop)

Using the given values, we can calculate the value of k:

k = -ln(1/1.5)/4

Substituting the value of k, we can calculate t_stop:

t_stop = -ln(1/1.5)/(-ln(1/1.5)/4)

Finally, we can calculate the distance traveled by the boat before coming to a stop:

d = 1.5/k * (1 - e^(-kt_stop))

Let's calculate the values:

k = -ln(1/1.5)/4 ≈ 0.1823

t_stop = -ln(1/1.5)/(0.1823) ≈ 4.772 seconds

d = 1.5/0.1823 * (1 - e^(-0.1823 * 4.772)) ≈ 6.153 meters

Therefore, the boat will come to a stop when its speed decreases to 1 cm/s after approximately 4.772 seconds. The boat can travel approximately 6.153 meters before coming to a stop.

Answer

The boat will come to a stop when its speed decreases to 1 cm/s after approximately 4.772 seconds. The boat can travel approximately 6.153 meters before coming to a stop.