A particle moves along the x-axis so that at time t, is greater than or equal to, 0t≥0 its position is given by x, left bracket, t, right bracket, equals, 5, t, to the power 4 , minus, 30, t, squared, .x(t)=5t 4 −30t 2 . Determine all intervals when the acceleration of the particle is positive.
Question
A particle moves along the x-axis so that at time t, is greater than or equal to, 0t≥0 its position is given by x, left bracket, t, right bracket, equals, 5, t, to the power 4 , minus, 30, t, squared, .x(t)=5t 4 −30t 2 . Determine all intervals when the acceleration of the particle is positive.
Solution
To find when the acceleration of the particle is positive, we first need to find the acceleration function. Acceleration is the second derivative of the position function.
The given position function is x(t) = 5t^4 - 30t^2.
First, let's find the velocity function, which is the first derivative of the position function.
v(t) = dx/dt = 20t^3 - 60t.
Next, let's find the acceleration function, which is the derivative of the velocity function.
a(t) = dv/dt = 60t^2 - 60.
We want to find when this acceleration function is positive, i.e., when a(t) > 0.
So, we solve the inequality 60t^2 - 60 > 0.
Divide through by 60 to simplify:
t^2 - 1 > 0.
This inequality is true when t < -1 or t > 1.
So, the acceleration of the particle is positive for t in the intervals (-∞, -1) and (1, ∞).
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