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The coordinates of a particle moving in a plane are given by x(t)=acos(pt) and y(t)=b sin(pt) where a,b(<a) and p are positive constants of appropriate dimensions. Then which of the following option is not true
- a)the distance travelled by the particle in the time interval t=0 to t=π/2p is a.
- b)the path of the particle is an ellipse
- c)the acceleration of the particle is always directed towards a focus
- d)the velocity and acceleration of the particle are normal to each other at t=π/2p
Correct answer is option 'A'. Can you explain this answer?
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The coordinates of a particle moving in a plane are given by x(t)=acos...
A, b are constants and p is a parameter. The parameter p determines the speed at which the particle moves.
To find the velocity vector of the particle, we need to differentiate x(t) and y(t) with respect to time.
dx/dt = -a p sin(pt)
dy/dt = b p cos(pt)
The velocity vector v(t) is given by v(t) = (dx/dt, dy/dt), so
v(t) = (-a p sin(pt), b p cos(pt))
To find the speed of the particle, we need to calculate 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))
= sqrt(a^2 p^2 sin^2(pt) + b^2 p^2 (1 - sin^2(pt))) (using the trigonometric identity sin^2(x) + cos^2(x) = 1)
= sqrt(a^2 p^2 sin^2(pt) + b^2 p^2 - b^2 p^2 sin^2(pt))
= sqrt(p^2 (a^2 sin^2(pt) + b^2 - b^2 sin^2(pt)))
= sqrt(p^2 (a^2 - (a^2 - b^2) sin^2(pt)))
= sqrt(p^2 (a^2 - c^2 sin^2(pt))), where c^2 = a^2 - b^2
Therefore, the speed of the particle is given by |v(t)| = p * sqrt(a^2 - c^2 sin^2(pt)).
Note: The speed of the particle is not constant, as it depends on the parameter p and the angle pt. The maximum speed occurs when sin^2(pt) = 1, which happens when pt = (2n + 1)π/2, where n is an integer. The minimum speed occurs when sin^2(pt) = 0, which happens when pt = nπ, where n is an integer.
To find the velocity vector of the particle, we need to differentiate x(t) and y(t) with respect to time.
dx/dt = -a p sin(pt)
dy/dt = b p cos(pt)
The velocity vector v(t) is given by v(t) = (dx/dt, dy/dt), so
v(t) = (-a p sin(pt), b p cos(pt))
To find the speed of the particle, we need to calculate 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))
= sqrt(a^2 p^2 sin^2(pt) + b^2 p^2 (1 - sin^2(pt))) (using the trigonometric identity sin^2(x) + cos^2(x) = 1)
= sqrt(a^2 p^2 sin^2(pt) + b^2 p^2 - b^2 p^2 sin^2(pt))
= sqrt(p^2 (a^2 sin^2(pt) + b^2 - b^2 sin^2(pt)))
= sqrt(p^2 (a^2 - (a^2 - b^2) sin^2(pt)))
= sqrt(p^2 (a^2 - c^2 sin^2(pt))), where c^2 = a^2 - b^2
Therefore, the speed of the particle is given by |v(t)| = p * sqrt(a^2 - c^2 sin^2(pt)).
Note: The speed of the particle is not constant, as it depends on the parameter p and the angle pt. The maximum speed occurs when sin^2(pt) = 1, which happens when pt = (2n + 1)π/2, where n is an integer. The minimum speed occurs when sin^2(pt) = 0, which happens when pt = nπ, where n is an integer.
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The coordinates of a particle moving in a plane are given by x(t)=acos(pt) and y(t)=b sin(pt) where a,b(<a) and p are positive constants of appropriate dimensions. Then which of the following option is not truea)the distance travelled by the particle in the time interval t=0 to t=π/2p is a.b)the path of the particle is an ellipsec)the acceleration of the particle is always directed towards a focusd)the velocity and acceleration of the particle are normal to each other at t=π/2pCorrect answer is option 'A'. Can you explain this answer?
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The coordinates of a particle moving in a plane are given by x(t)=acos(pt) and y(t)=b sin(pt) where a,b(<a) and p are positive constants of appropriate dimensions. Then which of the following option is not truea)the distance travelled by the particle in the time interval t=0 to t=π/2p is a.b)the path of the particle is an ellipsec)the acceleration of the particle is always directed towards a focusd)the velocity and acceleration of the particle are normal to each other at t=π/2pCorrect answer is option 'A'. 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)=acos(pt) and y(t)=b sin(pt) where a,b(<a) and p are positive constants of appropriate dimensions. Then which of the following option is not truea)the distance travelled by the particle in the time interval t=0 to t=π/2p is a.b)the path of the particle is an ellipsec)the acceleration of the particle is always directed towards a focusd)the velocity and acceleration of the particle are normal to each other at t=π/2pCorrect answer is option 'A'. 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)=acos(pt) and y(t)=b sin(pt) where a,b(<a) and p are positive constants of appropriate dimensions. Then which of the following option is not truea)the distance travelled by the particle in the time interval t=0 to t=π/2p is a.b)the path of the particle is an ellipsec)the acceleration of the particle is always directed towards a focusd)the velocity and acceleration of the particle are normal to each other at t=π/2pCorrect answer is option 'A'. Can you explain this answer?.
The coordinates of a particle moving in a plane are given by x(t)=acos(pt) and y(t)=b sin(pt) where a,b(<a) and p are positive constants of appropriate dimensions. Then which of the following option is not truea)the distance travelled by the particle in the time interval t=0 to t=π/2p is a.b)the path of the particle is an ellipsec)the acceleration of the particle is always directed towards a focusd)the velocity and acceleration of the particle are normal to each other at t=π/2pCorrect answer is option 'A'. 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)=acos(pt) and y(t)=b sin(pt) where a,b(<a) and p are positive constants of appropriate dimensions. Then which of the following option is not truea)the distance travelled by the particle in the time interval t=0 to t=π/2p is a.b)the path of the particle is an ellipsec)the acceleration of the particle is always directed towards a focusd)the velocity and acceleration of the particle are normal to each other at t=π/2pCorrect answer is option 'A'. 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)=acos(pt) and y(t)=b sin(pt) where a,b(<a) and p are positive constants of appropriate dimensions. Then which of the following option is not truea)the distance travelled by the particle in the time interval t=0 to t=π/2p is a.b)the path of the particle is an ellipsec)the acceleration of the particle is always directed towards a focusd)the velocity and acceleration of the particle are normal to each other at t=π/2pCorrect answer is option 'A'. Can you explain this answer?.
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