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the position x of a particle w.r.t. time t along x axis is given by x=9t^2 - t^3, what will be the position of this particle when it achieves max. speed along the. +x direction?1. 542. 813. 244. 32
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the position x of a particle w.r.t. time t along x axis is given by x=...
Given information:
The position of a particle with respect to time along the x-axis is given by the equation x = 9t^2 - t^3.
To find:
The position of the particle when it achieves maximum speed along the x-direction.
Solution:
To find the position of the particle when it achieves maximum speed, we need to find the velocity function and then locate the point where the velocity is maximum.
Step 1: Find the velocity function:
Velocity is the derivative of position with respect to time. Differentiating the given position function with respect to time will give us the velocity function.
The position function x = 9t^2 - t^3 can be rewritten as:
x = 9t^2 - t^3 = t^2(9 - t)
Now, let's differentiate x with respect to t to find the velocity function:
v = dx/dt = d/dt(t^2(9 - t))
v = 2t(9 - t) - t^2(1)
v = 18t - 3t^2 - t^2
v = 18t - 4t^2
Step 2: Find the maximum velocity:
To find the maximum velocity, we need to find the value of t for which the derivative of the velocity function is equal to zero.
dv/dt = d/dt(18t - 4t^2)
dv/dt = 18 - 8t
Setting dv/dt = 0, we have:
18 - 8t = 0
8t = 18
t = 18/8
t = 9/4
Step 3: Find the position when maximum velocity is achieved:
To find the position when maximum velocity is achieved, substitute the value of t = 9/4 into the position function x = 9t^2 - t^3.
x = 9(9/4)^2 - (9/4)^3
x = 9(81/16) - (729/64)
x = 729/16 - 729/64
x = 729/16 - 729/64 (LCM = 64)
x = (729*4 - 729)/64
x = 2916/64 - 729/64
x = 2187/64
Final Answer:
The position of the particle when it achieves maximum speed along the x-direction is 2187/64.
The position of a particle with respect to time along the x-axis is given by the equation x = 9t^2 - t^3.
To find:
The position of the particle when it achieves maximum speed along the x-direction.
Solution:
To find the position of the particle when it achieves maximum speed, we need to find the velocity function and then locate the point where the velocity is maximum.
Step 1: Find the velocity function:
Velocity is the derivative of position with respect to time. Differentiating the given position function with respect to time will give us the velocity function.
The position function x = 9t^2 - t^3 can be rewritten as:
x = 9t^2 - t^3 = t^2(9 - t)
Now, let's differentiate x with respect to t to find the velocity function:
v = dx/dt = d/dt(t^2(9 - t))
v = 2t(9 - t) - t^2(1)
v = 18t - 3t^2 - t^2
v = 18t - 4t^2
Step 2: Find the maximum velocity:
To find the maximum velocity, we need to find the value of t for which the derivative of the velocity function is equal to zero.
dv/dt = d/dt(18t - 4t^2)
dv/dt = 18 - 8t
Setting dv/dt = 0, we have:
18 - 8t = 0
8t = 18
t = 18/8
t = 9/4
Step 3: Find the position when maximum velocity is achieved:
To find the position when maximum velocity is achieved, substitute the value of t = 9/4 into the position function x = 9t^2 - t^3.
x = 9(9/4)^2 - (9/4)^3
x = 9(81/16) - (729/64)
x = 729/16 - 729/64
x = 729/16 - 729/64 (LCM = 64)
x = (729*4 - 729)/64
x = 2916/64 - 729/64
x = 2187/64
Final Answer:
The position of the particle when it achieves maximum speed along the x-direction is 2187/64.
Community Answer
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the position x of a particle w.r.t. time t along x axis is given by x=9t^2 - t^3, what will be the position of this particle when it achieves max. speed along the. +x direction?1. 542. 813. 244. 32
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