Given the position function s = 4t^3 − 24t^2 + 6t + 6.
find s and v when a = 0
(Where v is the velocity and a is the acceleration.).
v= ds/dt=12t^2-48t+6
a= dv/dt=24t-48=0 so t=2
find s, v when t=2
so i just have to sub t=2 in
v = 12t^2-48t+6 to get v and
s = 4t^3-24t^2+6t+6 to get s ?
To find the values of s and v when a is equal to 0, we need to differentiate the position function with respect to time to find the velocity function. Then, we can differentiate the velocity function with respect to time to find the acceleration function. Finally, we can substitute 0 into the acceleration function to find the values of s and v when a is equal to 0.
Step 1: Find the velocity function.
Differentiate the position function, s, with respect to time, t, to get the velocity function, v.
s = 4t^3 - 24t^2 + 6t + 6
To differentiate each term, we can use the power rule for differentiation:
The power rule states that if we have a term of the form ax^n, where a is a constant and n is any real number, then the derivative is given by d/dx(ax^n) = anx^(n-1).
Differentiating each term separately:
ds/dt = d/dt(4t^3) - d/dt(24t^2) + d/dt(6t) + d/dt(6)
ds/dt = 12t^2 - 48t + 6
Therefore, the velocity function is:
v = ds/dt = 12t^2 - 48t + 6
Step 2: Find the acceleration function.
Differentiate the velocity function, v, with respect to time, t, to get the acceleration function, a.
v = 12t^2 - 48t + 6
Differentiating each term separately:
dv/dt = d/dt(12t^2) - d/dt(48t) + d/dt(6)
dv/dt = 24t - 48
Therefore, the acceleration function is:
a = dv/dt = 24t - 48
Step 3: Substitute a = 0 to find s and v.
Set the acceleration function, a, equal to 0 and solve for t.
0 = 24t - 48
24t = 48
t = 48/24
t = 2
Now substitute t = 2 into the position function, s, and the velocity function, v, to find the values of s and v when a is equal to 0.
s = 4t^3 - 24t^2 + 6t + 6
s = 4(2)^3 - 24(2)^2 + 6(2) + 6
s = 32 - 96 + 12 + 6
s = -46
Therefore, when a is equal to 0, the position, s, is -46.
v = 12t^2 - 48t + 6
v = 12(2)^2 - 48(2) + 6
v = 48 - 96 + 6
v = -42
Therefore, when a is equal to 0, the velocity, v, is -42.