A particle starts at the origin and moves along the curve 4y = x^2 to the point (4, 4). If
the y-component of its acceleration is always equals 2 and the y-component of its velocity
is initially zero, find the x-component of its acceleration.
To find the x-component of the particle's acceleration, we need to determine its position function, velocity function, and acceleration function.
Given that the particle starts at the origin and moves along the curve 4y = x^2, we can find the position function by solving for y in terms of x:
4y = x^2
y = (1/4)x^2
Now, we can find the velocity function by taking the derivative of the position function with respect to time:
v = dy/dt
Since the y-component of its velocity is initially zero, we have:
v_y = 0
Differentiating y = (1/4)x^2 with respect to time, we get:
v_y = (1/2)x(dx/dt)
Since v_y = 0, we have:
0 = (1/2)x(dx/dt)
Since dx/dt represents the x-component of velocity, we can conclude:
dx/dt = 0
Now, to find the acceleration function, we take the derivative of the velocity function with respect to time:
a = dv/dt
Since the y-component of its acceleration is always 2, we have:
a_y = 2
Differentiating (1/2)x(dx/dt) with respect to time, we get:
a_y = 2 = (1/2)(d^2x/dt^2)
Therefore, we can conclude:
(d^2x/dt^2) = 4
The x-component of its acceleration is 4.