F=6kt

This force acts on a particle of mass "m" that moves along a line. If the particle starts from rest, determine its velocity and position at time t.

To determine the velocity and position of the particle at time t, we can use the equation F = mas, where F is the force, m is the mass, a is the acceleration, and s is the displacement.

Here, the force acting on the particle is given as F = 6kt. Since the particle starts from rest, its initial velocity (u) is zero.

To find the acceleration, we can rearrange the equation F = mas to solve for a:
a = F/m

Substituting the given values, we get:
a = (6kt)/m

Since the acceleration is the rate of change of velocity with respect to time, we can integrate the acceleration equation with respect to time to get the velocity equation:

∫a dt = ∫(6kt)/m dt

Integrating both sides gives us:
v = (3kt^2)/m + C

Here, v represents the velocity of the particle at time t, and C is the constant of integration. Since the particle starts from rest, the constant of integration, C, is zero.

Therefore, the velocity equation becomes:
v = (3kt^2)/m

To find the position of the particle, we can integrate the velocity equation with respect to time again:

∫v dt = ∫[(3kt^2)/m] dt

Integrating both sides gives us:
x = kt^3/m + C2

Here, x represents the position of the particle at time t, and C2 is the constant of integration.

Therefore, the position equation becomes:
x = kt^3/m

In summary:

Velocity at time t: v = (3kt^2)/m

Position at time t: x = kt^3/m

To determine the velocity and position of the particle at time t, we can integrate the force equation F = 6kt with respect to time.

Step 1: Solve for acceleration.
We know that force equals mass times acceleration (F = ma), so we can equate 6kt to ma:
6kt = ma

Dividing both sides by m, we can solve for acceleration:
a = 6k

Step 2: Integrate the acceleration equation to find velocity.
We can integrate the equation a = 6k with respect to time to find the velocity equation:
∫a dt = ∫6k dt
v = 6kt + C

Where C is the constant of integration. Since the particle starts from rest (initial velocity, v0 = 0), we can replace C with 0 to get:
v = 6kt

Step 3: Integrate the velocity equation to find the position.
Again, we can integrate the velocity equation v = 6kt with respect to time to find the position equation:
∫v dt = ∫6kt dt
x = 3kt^2 + C

Similarly, since the particle starts from rest (initial position, x0 = 0), we can replace C with 0 to get:
x = 3kt^2

Therefore, the velocity of the particle at time t is v = 6kt and the position of the particle at time t is x = 3kt^2.