How would I solve this:
-A particle moves along a line so that, at time t, its position is s(t)=8 sin2t.
a) For what values of t does the particle change direction?
b) What is the particle's maximum velocity?
a) The particle changes direction when the sine function changes sign, which occurs when t = (n + 1/2)π, where n is an integer.
b) The maximum velocity of the particle is 8, which occurs when the sine function is equal to 1, which occurs when t = nπ, where n is an integer.
To solve this problem, we need to determine when the particle changes direction and find its maximum velocity.
a) For what values of t does the particle change direction?
The particle changes direction when its velocity changes from positive to negative or from negative to positive. In other words, the particle changes direction when its velocity is equal to zero.
To find the velocity function, we need to differentiate the position function s(t) with respect to time t.
s(t) = 8 sin(2t)
Differentiating both sides of the equation with respect to t:
v(t) = ds(t)/dt = d(8 sin(2t))/dt
Using the chain rule, we have:
v(t) = 8 * d(sin(2t))/dt
Differentiating sin(2t) with respect to t:
v(t) = 8 * 2 cos(2t)
Now, set v(t) equal to 0 and solve for t:
8 * 2 cos(2t) = 0
cos(2t) = 0
To find values of t, for which cos(2t) = 0, we look for values of 2t where cos(2t) = 0. In other words, we need to find the times t when cos(2t) is equal to 0.
cos(2t) = 0 when 2t = π/2, 3π/2, 5π/2, ...
Dividing both sides by 2, we get:
t = π/4, 3π/4, 5π/4, ...
So, the particle changes direction at t = π/4, 3π/4, 5π/4, ...
b) What is the particle's maximum velocity?
To find the particle's maximum velocity, we need to find the maximum value of the absolute value of the velocity function v(t).
The velocity function is given by:
v(t) = 8 * 2 cos(2t)
To find the maximum value of |v(t)|, we need to find the maximum value of cos(2t), which is 1.
Therefore, the maximum value of |v(t)| is:
|v(t)| = 8 * 2 * 1 = 16
So, the particle's maximum velocity is 16.
To solve part (a) of the problem, we need to find the values of t for which the particle changes direction.
Let's start by understanding what it means for the particle to change direction. When the particle changes direction, its velocity changes from positive to negative, or from negative to positive. In other words, at the time the particle changes direction, its velocity is equal to zero.
So, we need to find when the velocity is equal to zero. Velocity is the derivative of position, so let's find the derivative of the given position function s(t):
s(t) = 8sin^2(t)
To find the derivative, we can use the chain rule. Let's differentiate each term one by one:
ds(t)/dt = d(8sin^2(t))/dt
= 16sin(t)cos(t)
Now, set this derivative equal to zero and solve for t to find the values where the particle changes direction:
16sin(t)cos(t) = 0
Using the fact that sin(t) = 0 when t = n*pi (where n is an integer) and cos(t) = 0 when t = (n + 1/2)*pi, we can find the values of t at which the particle changes direction:
For sin(t) = 0:
t = n*pi
For cos(t) = 0:
t = (n + 1/2)*pi
Therefore, the values of t at which the particle changes direction are t = n*pi and t = (n + 1/2)*pi, where n is an integer.
Now, let's move on to part (b) of the problem: finding the maximum velocity of the particle.
Velocity is given by the derivative of the position function s(t). We already found the derivative earlier:
ds(t)/dt = 16sin(t)cos(t)
To find the maximum velocity, we need to find when the derivative is equal to zero and determine whether it is a maximum or minimum point.
Setting the derivative equal to zero and solving for t gives us:
16sin(t)cos(t) = 0
Using the fact that sin(t) = 0 when t = n*pi and cos(t) = 0 when t = (n + 1/2)*pi, we can find the values of t at which the derivative is equal to zero:
For sin(t) = 0:
t = n*pi
For cos(t) = 0:
t = (n + 1/2)*pi
Now, we need to determine whether these values of t give us a maximum or minimum velocity. We can do this by looking at the second derivative of the position function:
d^2s(t)/dt^2 = -16sin(t)cos(t) + 16cos(t)sin(t)
= 0
Since the second derivative is equal to zero, we cannot determine whether the velocity is a maximum or minimum at these points.
However, we know that the position function s(t) = 8sin^2(t) is periodic, with a period of 2*pi. Therefore, we can conclude that the maximum velocity occurs at the midpoint of the period, which is t = pi.
Substituting t = pi into the derivative, we get:
ds(t)/dt = 16sin(pi)cos(pi)
= 16 * 0 * (-1)
= 0
Thus, the maximum velocity of the particle is 0 units per time.
In summary:
a) The particle changes direction at t = n*pi and t = (n + 1/2)*pi, where n is an integer.
b) The particle's maximum velocity is 0 units per time.