A particle is moving so that its displacement s (in metres) is given by s=10t−4t^3 where t is the time in seconds. Find the second derivative in order to find the equation for velocity. Find t when the particle is stationary (the velocity is zero)

To find the second derivative of the displacement function, we differentiate the equation twice with respect to time. Let's start by finding the first derivative: s = 10t - 4t^3 Taking the derivative of s with respect to t: ds/dt = d/dt(10t - 4t^3) = 10 - 12t^2 Now, we can find the second derivative by differentiating again: d^2s/dt^2 = d/dt(10 - 12t^2) = -24t The second derivative, d^2s/dt^2, represents the acceleration of the particle. To find the time when the particle is stationary (velocity is zero), we set the first derivative equal to zero: 10 - 12t^2 = 0 Solving this equation, we can rearrange it to: 12t^2 = 10 t^2 = 10/12 t^2 = 5/6 t = ±√(5/6) So, the particle is stationary at two different times: t = √(5/6) seconds and t = -√(5/6) seconds.