Question

A block of mass m is pulled by a rope along a horizontal surface with negligible friction. For time t> 0, the magnitude of the acceleration of the object as a function of time is given by the equation a = Ae determines the impulse exerted on the block by the rope during the time interval 0A) A∫_0^f e^-kt dt

B) mAl e^-kt dt
C) mAt_1⋅(e^−kt_1−e^0)
D) mAt_1⋅e^−kt_1

Answer 1

The impulse exerted on the block by the rope, you can use the definition of impulse, which is the integral of force with respect to time. By integrating the equation for acceleration and evaluating it from 0 to t1, you can determine the correct form of the impulse.

The impulse exerted on the block by the rope during the time interval 0 ≤ t ≤ t1, you can use the definition of impulse, which is the integral of force with respect to time. The equation given is for acceleration, but you can use Newton's second law (F = ma) to relate acceleration to force:

F = ma

The force is the derivative of momentum with respect to time:

F = dp/dt

Impulse is the change in momentum, so it is given by the integral of force with respect to time:

J = ∫ F dt

Now, substitute F = ma into the integral:

J = ∫ ma dt

Given a = Ae-kt, you can substitute this into the equation:

J = ∫ mAe-kt dt

Now, integrate with respect to time:

J = mA ∫ e-kt dt

Integrating e-kt with respect to t, you get -1/k e-kt. Therefore:

J = -mA/k e-kt

Now, evaluate this expression from 0 to t1:

J = -mA/k e-kt1 - (-mA/k e0)

Simplify:

J = (mA/k)(1 - e-kt1)

Now, let's compare this with the given options:

1. A) A∫0f e-kt dt - This is not the correct form.
2. B) mAt1 e-kt dt - This is not the correct form.
3. C) mAt1⋅(e-kt1 - e0) - This is the correct form.
4. D) mAt1⋅e-kt1 - This is not the correct form.

So, the correct answer is option C: J = mAt1⋅(e-kt1 - e0)

Answer 2

Impulse is the product of force and time. For a time-varying force, it is calculated by integrating the force over the time period. The impulse on the block with variable acceleration can be found using the given integral.

Step-by-step explanation:

The subject question pertains to the concept of impulse in physics. Impulse is defined as the change in momentum of an object when a force is applied over a period of time. The impulse can be calculated as the product of the average force and the time interval over which the force is applied (Impulse = Force × Time).

For the case where the force varies with time, the impulse can be found by integrating the force over the time interval. In the context of the given problem, since the acceleration a as a function of time is given by a = Ae-kt, the force exerted by the rope is F(t) = ma(t) = mAe-kt. The impulse I imparted on the block over a time interval from 0 to t1 is the integral of the force with respect to time:

I = ∫0t1F(t) dt = m∫0t1Ae-kt dt

This integral results in the following expression for impulse:

I = mAt1·(e−kt1−e0)