Question

Тело движется прямолинейно по закону s(t)=10t^2 - 2t^3. Найти скорость и ускорение движения тела в конце 2-ой секунды.

Answer 1

Нам дана функция его пройденного пути от времени. Скорость тела является производной этой функции:
v(t) = (10t^2-2t^3)' = 20t - 6t^2
Ускорение является производной скорости, или второй производной пути:
a(t) = (20t - 6t^2)' = 20 - 12t
Подставляя t=2, найдём значения:
v(2) = 20*2 - 6*2^2 = 40 - 24 = 16
a(2) = 20 - 12*2 = 20 - 24 = -4
Отрицательное ускорение означает, что движение равнозамедленное, т.е. происходит с торможением.

Answer 2

Given Information

The body is moving in a straight line according to the equation s(t) = 10t^2 - 2t^3.

Finding Velocity

To find the velocity of the body, we need to differentiate the equation s(t) with respect to time t. The velocity v(t) is given by the derivative of s(t) with respect to t.

Taking the derivative of s(t), we get:

v(t) = s'(t) = (10t^2 - 2t^3)' = 20t - 6t^2.

Now, we can find the velocity at the end of the 2nd second by substituting t = 2 into the equation for v(t):

v(2) = 20(2) - 6(2)^2 = 40 - 24 = 16.

Therefore, the velocity of the body at the end of the 2nd second is 16 units per second.

Finding Acceleration

To find the acceleration of the body, we need to differentiate the velocity equation v(t) with respect to time t. The acceleration a(t) is given by the derivative of v(t) with respect to t.

Taking the derivative of v(t), we get:

a(t) = v'(t) = (20t - 6t^2)' = 20 - 12t.

Now, we can find the acceleration at the end of the 2nd second by substituting t = 2 into the equation for a(t):

a(2) = 20 - 12(2) = 20 - 24 = -4.

Therefore, the acceleration of the body at the end of the 2nd second is -4 units per second squared.

Summary

In summary, the velocity of the body at the end of the 2nd second is 16 units per second, and the acceleration of the body at the end of the 2nd second is -4 units per second squared.

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