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The coordinates of a particle moving in a plane are given by x(t) = a cos (pt) and y (t) = b sin (pt) where a, b (< a) and p are positive constants of appropriate dimensions. Then
- 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 acceler ation of the particle is always directed towards a focus
- d)the distance travelled by the particle in time interval t = 0 tot =π/(2p) is a
Correct answer is option 'A,B,C'. Can you explain this answer?
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Most Upvoted Answer
The coordinates of a particle moving in a plane are given by x(t) = a ...
>0) and p (>0) are constants.
To find the velocity of the particle, we need to take the derivatives of x(t) and y(t) with respect to time t:
dx/dt = -a p sin (pt)
dy/dt = b p cos (pt)
The velocity of the particle is the vector (dx/dt, dy/dt), which can be written as:
v(t) = (-a p sin (pt), b p cos (pt))
To find the speed of the particle, we need to find the magnitude of the velocity vector:
|v(t)| = sqrt((-a p sin (pt))^2 + (b p cos (pt))^2)
= sqrt(a^2 p^2 sin^2 (pt) + b^2 p^2 cos^2 (pt))
= p sqrt(a^2 sin^2 (pt) + b^2 cos^2 (pt))
Since p, a, and b are constants, the speed of the particle depends only on t and the values of a and b:
speed = |v(t)| = p sqrt(a^2 sin^2 (pt) + b^2 cos^2 (pt))
To find the velocity of the particle, we need to take the derivatives of x(t) and y(t) with respect to time t:
dx/dt = -a p sin (pt)
dy/dt = b p cos (pt)
The velocity of the particle is the vector (dx/dt, dy/dt), which can be written as:
v(t) = (-a p sin (pt), b p cos (pt))
To find the speed of the particle, we need to find the magnitude of the velocity vector:
|v(t)| = sqrt((-a p sin (pt))^2 + (b p cos (pt))^2)
= sqrt(a^2 p^2 sin^2 (pt) + b^2 p^2 cos^2 (pt))
= p sqrt(a^2 sin^2 (pt) + b^2 cos^2 (pt))
Since p, a, and b are constants, the speed of the particle depends only on t and the values of a and b:
speed = |v(t)| = p sqrt(a^2 sin^2 (pt) + b^2 cos^2 (pt))
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The coordinates of a particle moving in a plane are given by x(t) = a ...
A and b are positive constants) and p (p is a positive constant) are parameters.
To find the trajectory of the particle, we can eliminate the parameter t by solving for t in terms of x and y using the trigonometric identities:
cos^2 (pt) + sin^2 (pt) = 1
cos (pt) = x/a
sin (pt) = y/b
Squaring both equations and adding them, we get:
(x/a)^2 + (y/b)^2 = 1
This is the equation of an ellipse centered at the origin with semi-major axis a and semi-minor axis b. Thus, the trajectory of the particle is an ellipse centered at the origin and oriented along the x and y axes. The particle moves counterclockwise around the ellipse at a constant speed determined by the parameter p.
To find the trajectory of the particle, we can eliminate the parameter t by solving for t in terms of x and y using the trigonometric identities:
cos^2 (pt) + sin^2 (pt) = 1
cos (pt) = x/a
sin (pt) = y/b
Squaring both equations and adding them, we get:
(x/a)^2 + (y/b)^2 = 1
This is the equation of an ellipse centered at the origin with semi-major axis a and semi-minor axis b. Thus, the trajectory of the particle is an ellipse centered at the origin and oriented along the x and y axes. The particle moves counterclockwise around the ellipse at a constant speed determined by the parameter p.
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The coordinates of a particle moving in a plane are given by x(t) = a cos (pt) and y (t) = b sin (pt) where a, b (< a) and p are positive constants of appropriate dimensions. Thena)the path of the particle is an ellipseb)the velocity and acceleration of the particle are normal to each other at t =π/(2p)c)the acceler ation of the particle is always directed towards a focusd)the distance travelled by the particle in time interval t = 0 tot =π/(2p) is aCorrect answer is option 'A,B,C'. Can you explain this answer?
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The coordinates of a particle moving in a plane are given by x(t) = a cos (pt) and y (t) = b sin (pt) where a, b (< a) and p are positive constants of appropriate dimensions. Thena)the path of the particle is an ellipseb)the velocity and acceleration of the particle are normal to each other at t =π/(2p)c)the acceler ation of the particle is always directed towards a focusd)the distance travelled by the particle in time interval t = 0 tot =π/(2p) is aCorrect answer is option 'A,B,C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The coordinates of a particle moving in a plane are given by x(t) = a cos (pt) and y (t) = b sin (pt) where a, b (< a) and p are positive constants of appropriate dimensions. Thena)the path of the particle is an ellipseb)the velocity and acceleration of the particle are normal to each other at t =π/(2p)c)the acceler ation of the particle is always directed towards a focusd)the distance travelled by the particle in time interval t = 0 tot =π/(2p) is aCorrect answer is option 'A,B,C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The coordinates of a particle moving in a plane are given by x(t) = a cos (pt) and y (t) = b sin (pt) where a, b (< a) and p are positive constants of appropriate dimensions. Thena)the path of the particle is an ellipseb)the velocity and acceleration of the particle are normal to each other at t =π/(2p)c)the acceler ation of the particle is always directed towards a focusd)the distance travelled by the particle in time interval t = 0 tot =π/(2p) is aCorrect answer is option 'A,B,C'. Can you explain this answer?.
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