An object is moving along the x-axis. Its position at any time is given by x(t) = 3cos(pi * t) + 2. Find its acceleration.
v = dx/dt = -3 pi sin (pi * t)
a = d^2x/dt^2 = -3 pi^2 cos (pi * t)
Well, well, well, let's find that acceleration of our moving object along the x-axis. To do that, we'll take the second derivative of its position function. Get ready for some mathematical magic!
The position function is x(t) = 3cos(pi * t) + 2. Now, the first derivative (or the velocity) will be found by differentiating the function with respect to time. Since the derivative of cosine is negative sine, the velocity function will be v(t) = -3pi * sin(pi * t).
Now, it's time for the grand finale, the second derivative! We'll differentiate the velocity function with respect to time, and we get a(t) = -3pi^2 * cos(pi * t).
Ta-da! There you have it, the acceleration function a(t) = -3pi^2 * cos(pi * t). I hope this answer wasn't too "derivative" for you. Remember, I'm always here to add a little humor to your math questions!
To find the acceleration of an object, we need to determine its second derivative with respect to time.
Given that the position of the object is given by the equation x(t) = 3cos(pi * t) + 2, we can find the velocity by taking the first derivative of x(t) with respect to time.
Let's start by finding the velocity:
v(t) = dx(t)/dt
To find the derivative of x(t), we can differentiate each term separately, using the chain rule for the first term:
dx(t)/dt = d(3cos(pi * t))/dt + d(2)/dt
Differentiating each term:
dx(t)/dt = -3(pi)sin(pi * t) + 0
Simplifying, we have:
dx(t)/dt = -3(pi)sin(pi * t)
Now, let's find the acceleration by taking the derivative of the velocity:
a(t) = dv(t)/dt
To find the derivative of v(t), we can differentiate each term separately, using the chain rule for the first term:
dv(t)/dt = d(-3(pi)sin(pi * t))/dt
Differentiating each term:
dv(t)/dt = -3(pi)(d(sin(pi * t))/dt)
Taking the derivative of sin(pi * t):
dv(t)/dt = -3(pi)(pi)cos(pi * t)
Simplifying, we have:
dv(t)/dt = -3(pi^2)cos(pi * t)
Therefore, the acceleration of the object is given by a(t) = -3(pi^2)cos(pi * t).
To find the acceleration of an object, we need to take the second derivative of the position function with respect to time.
Given that the position function is x(t) = 3cos(pi * t) + 2, we can find its derivative by applying the chain rule:
x'(t) = -3(pi) * sin(pi * t)
Next, we take the derivative of x'(t) with respect to time to find the acceleration:
x''(t) = [d/dt] (-3(pi) * sin(pi * t))
To differentiate this expression, we use the chain rule again:
x''(t) = -3(pi)^2 * cos(pi * t)
Therefore, the acceleration of the object is given by a(t) = -3(pi)^2 * cos(pi * t).