The one-dimensional displacement, s meters of a particle, after t seconds, is given by the function s=t(t-4)^2.
(i) when does the particle have zero acceleration?
(ii) where is the particle at this time and what is it doing?
s = t(t^2 - 8t + 16)
= t^3 - 8t^2 + 16t
v = 3t^2 - 16t + 16
a = 6t - 16
for zero of a,
6t-16=0
t = 16/6 = 8/3 seconds
when t = 8/3
v = 3(64/9) - 16(8/3) + 16
= -16/3
s = (8/3)(8/3 - 4)^2 = 128/27
work these answers in with your concluding statements
To find when the particle has zero acceleration, we need to determine the time when the particle's acceleration is equal to zero.
Acceleration can be found by taking the second derivative of the displacement function. So we need to find the second derivative of the function s(t).
(i) Let's calculate the acceleration by finding the second derivative of s(t).
First, find the derivative of s(t) with respect to t, which gives us the velocity function v(t):
v(t) = ds(t)/dt = d/dt [t(t-4)^2]
We can use the product rule to find this derivative:
v(t) = t(d/dt[(t-4)^2]) + (t-4)^2(d/dt[t])
= t(2(t-4)(1)) + (t-4)^2(1)
= 2t(t-4) + (t-4)^2
Now, find the second derivative of s(t) to get the acceleration function a(t):
a(t) = dv(t)/dt = d/dt [2t(t-4) + (t-4)^2]
= 2(d/dt[t(t-4)]) + d/dt[(t-4)^2]
= 2(t-4) + 2(t-4)
= 4(t-4)
To find when the particle has zero acceleration, we need to solve the equation a(t) = 4(t-4) = 0.
4(t-4) = 0
t-4 = 0
t = 4
Therefore, the particle has zero acceleration at t = 4 seconds.
(ii) Now let's determine the position and behavior of the particle at t = 4 seconds.
To find the position at t = 4 seconds, we substitute t = 4 into the displacement function s(t):
s(4) = 4(4-4)^2
= 4(0)^2
= 0
Therefore, at t = 4 seconds, the particle is located at s = 0 meters.
To understand what the particle is doing, we can analyze the behavior of the displacement function s(t) in the vicinity of t = 4.
Looking at the function s(t) = t(t-4)^2, we observe that the squared term (t-4)^2 represents a positive quantity, which means that the particle's displacement is nonzero for t ≠ 4.
However, at t = 4 seconds, the particle's displacement is zero (s = 0), indicating that the particle is momentarily stationary at this point in time.
In summary, at t = 4 seconds, the particle is located at s = 0 meters, and it is momentarily stationary.