IIT JEE , AIEEE Forum - Physics , Mechanics

elessar_iitkgp

To know the nature of the path of the particle we must know the law of variation of position vector with time, for the path is but the locus of the end point of the position vector
If you are given velocity, then
dr/dt = v
_r0

^r dr = _t0

^t vdt
r = r₀ + _t0

^t vdt
where r₀ is the position vector of the particle at time t. , ie, r₀= r(t₀)
Now, until and unless r₀ is given, you can't determine r. So with velocity alone, the law of variation of position vector can't be found. Its position vector at a certain instant t₀ must be known before hand.

Similar remarks apply to acceleration. If you start integrating from a then you will need the velocity v₀ at certain time t₀ as well as the position vector r₀ at that time.

Hence the option (D) is correct

@feynamnn,
The nature of the path can only be analyzed if you have the equation of the curve in any coordinate form .Most commonly its the Cartesian Coordinate form, ie, once you know the function r = r(t), you can generate three equations of the form
x = x(t)
y=y(t)
z=z(t)
Eliminate t from these three and you get the equation of the path. From this equation you can then analyze the nature of the path, ie, intervals of increse/decrease, curvature and all sorts of things

Similar remarks apply for any other coordinate system such as plane polar, spherical polar, cylindrical polar, or a generalized coordinate system.

@viv
Yes, ultimately it is the force applied that governs the path. It is apparent once you start from acceleration as acceleration itself is caused by force.

Given the position r, one can determine the velocity v and acceleration a of the particle without any further information using
v = dr/dt
a=dv/dt
However, the three kinematics equation are just the solution to the reverse problem:
"To find the velocity and position of a particle if the acceleration is given"
One can extend this to
"To find the velocity and position of a particle if the force acting on the particle is given"
Now, it turns out that the solutions to this problem requires two initial conditions, namely, the position and velocity at a certain moment of time t₀, ie, v₀ and r₀

Ask & Discuss Questions with Community & Experts