Hard
Question
The coordinate of a particle moving in a plane are given by x (t) = a cos ( pt) and y (t) = bsin ( pt) , where a, b (< a) and p are positive onstants of appropriate dimensions. Then
Options
A.the path of the particle is an ellipse
B.the velocity and acceleration of the particle are normal to each other at t = π/2p
C.the acceleration of the particle is always directed towards a focus
D.the distance travelled by the particle in time interval t = 0 to t = π/2p is a.
Solution
x = a cos ( pt) ⇒ cos(pt) = 
......(i)
y = b sin ( pt) ⇒ sin (pt) =
......(ii)
Squaring and adding Eqs. (i) and (ii), we get
= 1
Therefore, path of the particle is an ellipse. Hence option (a) is correct.From the given equation we can find
= vx = - ap sin pt
= ax = - a2 cos pt
= vy = bp cos pt and
ay = - bp2 sin pt
At time t = π/2p or pt = π/2
ax and vy become zero (because cos π/2 = 0)
only vx and ay are left.
or we can sat that velocity is along negative x-axis and acceleration along y-axis
Hence, at t = π/2p velocity and acceleration of the particle are normal to each other. So option (b) is also correct.
At t = t, position of the particle
and acceleration of the particle is
Therefore, acceleration of the particle is always directed towards origin.
Hence, option (c) is also corrent.
At t = 0 particle is at (a, 0) and t = π/2p particle is at (0, b). Therefore, the distance covered is one-fouth of the
......(i)y = b sin ( pt) ⇒ sin (pt) =
......(ii)Squaring and adding Eqs. (i) and (ii), we get
= 1Therefore, path of the particle is an ellipse. Hence option (a) is correct.From the given equation we can find
= vx = - ap sin pt = ax = - a2 cos pt = vy = bp cos pt and ay = - bp2 sin pt At time t = π/2p or pt = π/2
ax and vy become zero (because cos π/2 = 0)
only vx and ay are left.
or we can sat that velocity is along negative x-axis and acceleration along y-axis
Hence, at t = π/2p velocity and acceleration of the particle are normal to each other. So option (b) is also correct.
At t = t, position of the particle
and acceleration of the particle is
Therefore, acceleration of the particle is always directed towards origin.
Hence, option (c) is also corrent.
At t = 0 particle is at (a, 0) and t = π/2p particle is at (0, b). Therefore, the distance covered is one-fouth of the
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