An object of mass m moves in an elliptical orbit. The position of this object are vector vectors r ⃗= bsinwt + 2bcoswt j (i and j are unit vectors) given as. According to this; a) Determine the velocity, velocity and acceleration of the object. b) What is the angle between the velocity and acceleration of the object when t = π / 2w?
Answers
Answer:
Instantaneous Acceleration
In addition to obtaining the displacement and velocity vectors of an object in motion, we often want to know its acceleration vector at any point in time along its trajectory. This acceleration vector is the instantaneous acceleration and it can be obtained from the derivative with respect to time of the velocity function, as we have seen in a previous chapter. The only difference in two or three dimensions is that these are now vector quantities. Taking the derivative with respect to time →v(t),v→(t), we find
→a(t)=limt→0→v(t+Δt)−→v(t)Δt=d→v(t)dt.a→(t)=limt→0v→(t+Δt)−v→(t)Δt=dv→(t)dt.
The acceleration in terms of components is
→a(t)=dvx(t)dt^i+dvy(t)dt^j+dvz(t)dt^k.a→(t)=dvx(t)dti^+dvy(t)dtj^+dvz(t)dtk^.
Also, since the velocity is the derivative of the position function, we can write the acceleration in terms of the second derivative of the position function:
→a(t)=d
Answer:
For vector calculus, we make the same definition. Let r(t) be a differentiable vector valued function representing the position vector of a particle at time t. Then the velocity vector is the derivative of the pose