Question

In which x is the position of the particle, v is its velocity and k is a positive constant. If the equation of motion is given by x = k * t² + v * t, what is the acceleration of the particle?

1) 2k + v
2) 2k
3) 2v
4) k + v

Answer 1

The acceleration of the particle, given the equation of motion x = kt² + vt, is 2k, which corresponds to option 2).

Step-by-step explanation:

The equation of motion for a particle is given by x = kt² + vt, where x is the position, t is time, v is the initial velocity and k is a positive constant. To find the acceleration of the particle, we need to take the second derivative of the position function with respect to time.

The first derivative (dx/dt) gives the velocity function, which is 2kt + v. Taking another derivative of this velocity function with respect to time gives the acceleration function (d²x/dt²). Hence, the acceleration is 2k, which is option 2).

Acceleration does not depend on the initial velocity v, as velocity is the first derivative of position and acceleration is the second derivative; changes in velocity that are linear with respect to time (such as the term vt) will not affect the acceleration.