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The co-ordinates of a moving particle at a time t, are given by, x = 5 sin 10 t, y = 5 cos 10t. The speed of the particle is : Q.4 An object is moving along the x axis with position as a function of time given by x = x(t). Point O is a?
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?The co-ordinates of a moving particle at a time t, are given by, x = ...
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Introduction:
The problem provides the coordinates of a moving particle at a given time. We are required to find the speed of the particle. Additionally, we need to determine the nature of point O based on the object's motion along the x-axis.
Given Information:
The coordinates of the particle at time 't' are given as:
x = 5 sin 10t
y = 5 cos 10t
Calculating Speed:
To find the speed of the particle, we need to calculate the magnitude of its velocity. The velocity can be obtained by differentiating the position equations with respect to time, t.
Given:
x = 5 sin 10t
Differentiating x with respect to t:
v_x = dx/dt = 5 * (d/dt) (sin 10t)
v_x = 5 * 10 cos 10t
v_x = 50 cos 10t
Similarly, differentiating y with respect to t:
v_y = dy/dt = 5 * (d/dt) (cos 10t)
v_y = 5 * (-10) sin 10t
v_y = -50 sin 10t
The magnitude of velocity, v, can be calculated using the Pythagorean theorem:
v = √(v_x^2 + v_y^2)
v = √((50 cos 10t)^2 + (-50 sin 10t)^2)
v = √(2500 cos^2 10t + 2500 sin^2 10t)
v = √2500 (cos^2 10t + sin^2 10t)
v = √2500
v = 50
Hence, the speed of the particle is a constant value of 50.
Determining Point O:
Point O can be determined based on the object's motion along the x-axis. From the given position equation x = 5 sin 10t, we can observe the following:
- When t = 0, x = 5 sin(0) = 0
- When t = π/20, x = 5 sin(10 * π/20) = 5 sin(π/2) = 5
- When t = π/10, x = 5 sin(10 * π/10) = 5 sin(π) = 0
- When t = 3π/20, x = 5 sin(10 * 3π/20) = 5 sin(3π/2) = -5
From the above observations, we can conclude that point O lies at x = 0. Therefore, point O is the origin (0, 0) on the x-axis.
Summary:
- The speed of the particle is a constant value of 50.
- Point O is the origin (0, 0) on the x-axis.
The problem provides the coordinates of a moving particle at a given time. We are required to find the speed of the particle. Additionally, we need to determine the nature of point O based on the object's motion along the x-axis.
Given Information:
The coordinates of the particle at time 't' are given as:
x = 5 sin 10t
y = 5 cos 10t
Calculating Speed:
To find the speed of the particle, we need to calculate the magnitude of its velocity. The velocity can be obtained by differentiating the position equations with respect to time, t.
Given:
x = 5 sin 10t
Differentiating x with respect to t:
v_x = dx/dt = 5 * (d/dt) (sin 10t)
v_x = 5 * 10 cos 10t
v_x = 50 cos 10t
Similarly, differentiating y with respect to t:
v_y = dy/dt = 5 * (d/dt) (cos 10t)
v_y = 5 * (-10) sin 10t
v_y = -50 sin 10t
The magnitude of velocity, v, can be calculated using the Pythagorean theorem:
v = √(v_x^2 + v_y^2)
v = √((50 cos 10t)^2 + (-50 sin 10t)^2)
v = √(2500 cos^2 10t + 2500 sin^2 10t)
v = √2500 (cos^2 10t + sin^2 10t)
v = √2500
v = 50
Hence, the speed of the particle is a constant value of 50.
Determining Point O:
Point O can be determined based on the object's motion along the x-axis. From the given position equation x = 5 sin 10t, we can observe the following:
- When t = 0, x = 5 sin(0) = 0
- When t = π/20, x = 5 sin(10 * π/20) = 5 sin(π/2) = 5
- When t = π/10, x = 5 sin(10 * π/10) = 5 sin(π) = 0
- When t = 3π/20, x = 5 sin(10 * 3π/20) = 5 sin(3π/2) = -5
From the above observations, we can conclude that point O lies at x = 0. Therefore, point O is the origin (0, 0) on the x-axis.
Summary:
- The speed of the particle is a constant value of 50.
- Point O is the origin (0, 0) on the x-axis.
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?The co-ordinates of a moving particle at a time t, are given by, x = 5 sin 10 t, y = 5 cos 10t. The speed of the particle is : Q.4 An object is moving along the x axis with position as a function of time given by x = x(t). Point O is a?
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?The co-ordinates of a moving particle at a time t, are given by, x = 5 sin 10 t, y = 5 cos 10t. The speed of the particle is : Q.4 An object is moving along the x axis with position as a function of time given by x = x(t). Point O is a? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about ?The co-ordinates of a moving particle at a time t, are given by, x = 5 sin 10 t, y = 5 cos 10t. The speed of the particle is : Q.4 An object is moving along the x axis with position as a function of time given by x = x(t). Point O is a? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ?The co-ordinates of a moving particle at a time t, are given by, x = 5 sin 10 t, y = 5 cos 10t. The speed of the particle is : Q.4 An object is moving along the x axis with position as a function of time given by x = x(t). Point O is a?.
?The co-ordinates of a moving particle at a time t, are given by, x = 5 sin 10 t, y = 5 cos 10t. The speed of the particle is : Q.4 An object is moving along the x axis with position as a function of time given by x = x(t). Point O is a? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about ?The co-ordinates of a moving particle at a time t, are given by, x = 5 sin 10 t, y = 5 cos 10t. The speed of the particle is : Q.4 An object is moving along the x axis with position as a function of time given by x = x(t). Point O is a? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ?The co-ordinates of a moving particle at a time t, are given by, x = 5 sin 10 t, y = 5 cos 10t. The speed of the particle is : Q.4 An object is moving along the x axis with position as a function of time given by x = x(t). Point O is a?.
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