A particle moves a distance 'x' in time 't' according to equation x=(t...
Acceleration is the rate of change of velocity with respect to time. In order to determine the acceleration of the particle, we need to find the velocity function first.
1. Finding the velocity function:
We are given the equation for the distance traveled by the particle, x=(t^5)^-1. To find the velocity function, we need to differentiate x with respect to time (t).
Taking the derivative of x with respect to t, we can use the chain rule:
dx/dt = d/dt((t^5)^-1)
= -1(t^5)^-2 * d/dt(t^5)
= -1(t^5)^-2 * 5t^4
= -5t^4/(t^5)^2
= -5t^4/t^10
= -5/t^6
Therefore, the velocity function v(t) = dx/dt = -5/t^6.
2. Finding the acceleration function:
To find the acceleration function, we need to differentiate the velocity function with respect to time (t).
Taking the derivative of v(t) with respect to t:
dv/dt = d/dt(-5/t^6)
= 5(6/t^7)
= 30/t^7
Therefore, the acceleration function a(t) = dv/dt = 30/t^7.
3. Determining the relationship between acceleration and velocity:
Now, let's examine the relationship between acceleration and velocity. We have the velocity function v(t) = -5/t^6 and the acceleration function a(t) = 30/t^7.
Let's substitute the velocity function into the acceleration function:
a(t) = 30/t^7
= 30/((-5/t^6)^7)
= 30/((-5)^7/t^(6*7))
= 30/((-5)^7/t^42)
= 30/(5^7/t^42)
= (30*t^42)/(5^7)
From the above equation, we can see that the acceleration is proportional to (velocity)^3/2. Therefore, the correct option is 1.) (velocity)^3/2.
A particle moves a distance 'x' in time 't' according to equation x=(t...
x=( t+5)^-1 . v= dx/dt = -- ( t+5)^ -2. tht means (t+5)= --v ^ -1/2 . now, a= dv/dt = 2 ( t+5) ^ -3. , a= 2( - v ^ -1/2) ^ -3, a = -- 2 v ^ 3/2, tht is a is prop to vel ^3/2
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