Question

Путь, проходимый телом массой m = 2,0 кг, описывается уравнением S = A – Bt + Ct2 – Dt3 (t – время, A = 1,5 м, B = 6,0 м/c, С = 3,5 м/с2 , D = 3,0 м/с3 ). Какую мощность N развивает действующая на тело сила в момент времени t = 1,5 c ? Ответ: N = 0,63 кВт.

Answer 1

Дано:

m = 2 кг

S(t) = 1,5 - 6,0*t + 3,5*t² - 3,0*t³

t = 1,5 c

__________

N - ?

1)

Скорость - первая производная от пути по времени:

v(t) = S' (t) = - 6,0 +7,0*t - 9,0*t²

v(1,5) = -6,0 +7,0*1,5 - 9*1,5² ≈ - 15,75 м/с  (тело движется против положительного направления оси OX)

2)

Ускорение - первая производная от скорости:

a(t) = v'(t) = 7 - 18*t

a(1,5) = 7 - 18*1,5 = - 20 м/с²

3)

Сила:

F = m*a = 2,0*(-20) = - 40 Н

4)

Мощность:

N = F*v = - 40*(-15,75) =  630 Вт   или

N = 0,63 кВт

Answer 2

Calculation of Power

To calculate the power developed by the force acting on the body at a specific time, we need to find the derivative of the displacement equation with respect to time. The power is then given by the product of the force and the velocity of the body at that time.

The displacement equation is given as:

S = A - Bt + Ct^2 - Dt^3

where: - S is the displacement of the body, - A is the initial displacement, - B is the coefficient of the linear term, - C is the coefficient of the quadratic term, - D is the coefficient of the cubic term, and - t is the time.

To find the velocity of the body, we differentiate the displacement equation with respect to time:

V = dS/dt = -B + 2Ct - 3Dt^2

where V is the velocity of the body.

The power developed by the force acting on the body is given by:

N = F * V

where N is the power, and F is the force acting on the body.

To find the power at a specific time t, we substitute the given values into the equations and calculate the result.

Calculation of Power at t = 1.5 s

Given: - m = 2.0 kg (mass of the body) - A = 1.5 m - B = 6.0 m/s - C = 3.5 m/s^2 - D = 3.0 m/s^3 - t = 1.5 s

Substituting these values into the equations, we can calculate the power:

V = -B + 2Ct - 3Dt^2

V = -6.0 + 2(3.5)(1.5) - 3(3.0)(1.5)^2

V = -6.0 + 10.5 - 20.25

V = -15.75 m/s

Now, we need to calculate the force acting on the body. The force can be found using Newton's second law:

F = ma

where F is the force, m is the mass of the body, and a is the acceleration.

The acceleration can be found by differentiating the velocity equation with respect to time:

a = dV/dt = 2C - 6Dt

Substituting the given values:

a = 2(3.5) - 6(3.0)(1.5)

a = 7.0 - 27.0

a = -20.0 m/s^2

Now, we can calculate the force:

F = ma = (2.0 kg)(-20.0 m/s^2)

F = -40.0 N

Finally, we can calculate the power:

N = F * V = (-40.0 N)(-15.75 m/s)

N = 630 W

Converting the power to kilowatts:

N = 0.63 kW

Therefore, the power developed by the force acting on the body at t = 1.5 s is 0.63 kW.

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