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A particle is acted on by a force parallel to y axis whose acceleration is Lambda why and is initially projected with a velocity a under root Lambda parallel to x-axis at a point where Y is equal to a. Prove that it will describe the category Y is is equal to A cos hyperbolic X upon a?
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A particle is acted on by a force parallel to y axis whose acceleratio...
Problem Statement: A particle is acted on by a force parallel to the y-axis whose acceleration is Lambda and is initially projected with a velocity a√Lambda parallel to the x-axis at a point where Y is equal to a. Prove that it will describe the category Y is equal to A cosh(X/a).
Solution:
The given problem can be solved using the following steps:
Equation of Motion:
Given that the force acting on the particle is parallel to the y-axis and its acceleration is Lambda. Therefore, the equation of motion of the particle can be written as:
Integrating both sides with respect to t, we get:
Integrating both sides again with respect to t, we get:
Applying the initial conditions, we get:
Therefore, the equation of motion of the particle is:
Equation of Path:
The particle is initially projected with a velocity a√Lambda parallel to the x-axis. Therefore, the initial velocity of the particle can be written as:
Integrating both sides with respect to t, we get:
Substituting this value of t in the equation of motion of the particle, we get:
Simplifying the above equation, we get:
where A = a/2 and C3 = 0.
Therefore, the path of the particle is given by the equation Y = A cosh(X/a).
Hence, it is proved that the particle will describe the category Y is equal to A cosh(X/a).
Solution:
The given problem can be solved using the following steps:
- Find the equation of motion of the particle
- Derive the equation of the path of the particle
Equation of Motion:
Given that the force acting on the particle is parallel to the y-axis and its acceleration is Lambda. Therefore, the equation of motion of the particle can be written as:
d^2y/dt^2 = Lambda
Integrating both sides with respect to t, we get:
dy/dt = Lambda*t + C1
Integrating both sides again with respect to t, we get:
y = 1/2*Lambda*t^2 + C1*t + C2
Applying the initial conditions, we get:
a = C2
0 = Lambda*a + C1
Therefore, the equation of motion of the particle is:
y = 1/2*Lambda*t^2 - 1/2*Lambda*a^2
Equation of Path:
The particle is initially projected with a velocity a√Lambda parallel to the x-axis. Therefore, the initial velocity of the particle can be written as:
dx/dt = a*sqrt(Lambda)
Integrating both sides with respect to t, we get:
x = a*sqrt(Lambda)*t + C3
t = (x - C3)/a*sqrt(Lambda)
Substituting this value of t in the equation of motion of the particle, we get:
y = 1/2*Lambda*((x - C3)/a*sqrt(Lambda))^2 - 1/2*Lambda*a^2
Simplifying the above equation, we get:
y = A*cosh(X/a)
where A = a/2 and C3 = 0.
Therefore, the path of the particle is given by the equation Y = A cosh(X/a).
Hence, it is proved that the particle will describe the category Y is equal to A cosh(X/a).
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A particle is acted on by a force parallel to y axis whose acceleration is Lambda why and is initially projected with a velocity a under root Lambda parallel to x-axis at a point where Y is equal to a. Prove that it will describe the category Y is is equal to A cos hyperbolic X upon a?
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A particle is acted on by a force parallel to y axis whose acceleration is Lambda why and is initially projected with a velocity a under root Lambda parallel to x-axis at a point where Y is equal to a. Prove that it will describe the category Y is is equal to A cos hyperbolic X upon a? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about A particle is acted on by a force parallel to y axis whose acceleration is Lambda why and is initially projected with a velocity a under root Lambda parallel to x-axis at a point where Y is equal to a. Prove that it will describe the category Y is is equal to A cos hyperbolic X upon a? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle is acted on by a force parallel to y axis whose acceleration is Lambda why and is initially projected with a velocity a under root Lambda parallel to x-axis at a point where Y is equal to a. Prove that it will describe the category Y is is equal to A cos hyperbolic X upon a?.
A particle is acted on by a force parallel to y axis whose acceleration is Lambda why and is initially projected with a velocity a under root Lambda parallel to x-axis at a point where Y is equal to a. Prove that it will describe the category Y is is equal to A cos hyperbolic X upon a? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about A particle is acted on by a force parallel to y axis whose acceleration is Lambda why and is initially projected with a velocity a under root Lambda parallel to x-axis at a point where Y is equal to a. Prove that it will describe the category Y is is equal to A cos hyperbolic X upon a? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A particle is acted on by a force parallel to y axis whose acceleration is Lambda why and is initially projected with a velocity a under root Lambda parallel to x-axis at a point where Y is equal to a. Prove that it will describe the category Y is is equal to A cos hyperbolic X upon a?.
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