Question

a particle of mass m starts from rest at position x=0 and time t=0. it moves along the positive x axis under the influence of a single force F=bt, where b is a constant.the velocity of the particle is given by

Answer 1

The velocity (v) of the particle is v = bt / 2m, and the position (x) is x = bt^3 / 6m, where b is a constant force and m is the mass, starting from rest at x=0 and t=0.

Step-by-step explanation:

The velocity of the particle can be determined by integrating the force function with respect to time.

Starting from rest, the velocity function can be written as v(t) = ∫ F(t) dt = ∫ bt dt = 0.5bt^2 + C, where C is the integration constant.

Since the particle starts at x = 0 and t = 0, we can determine the value of C as 0, which gives us v(t) = 0.5bt^2.

Therefore, the velocity of the particle is given by v(t) = 0.5bt^2.

Answer 2

To find a particle's acceleration and position from a velocity-time function, differentiate to get the acceleration and integrate to obtain the position for specified times using the given initial conditions.

Step-by-step explanation:

The given question involves finding the acceleration and position of a particle at specific times, given its velocity-time relationship. For a velocity function v(t) = A + Bt⁻¹, where A and B are constants, the acceleration at any time t can be obtained by differentiating the velocity function with respect to time. Given that A = 2 m/s and B = 0.25 m, the acceleration a(t) would be the negative derivative of B with respect to t squared (a(t) = -Bt⁻²).

To find the position x(t), we integrate the velocity function with respect to time. Since initial conditions are given (x(t = 1 s) = 0), we can calculate the definite integral from 1 second to the desired time to find the position at that specific time.

For example:
At t = 2.0 s, acceleration a(2) = -0.25 / 2² = -0.0625 m/s², and the position x(2) can be found via integration from t = 1 to t = 2 of the velocity function. Similarly, for t = 5.0 s, one would find acceleration a(5) = -0.25 / 5² = -0.01 m/s² and then calculate position x(5) with the same method.