A particle's position on the x-axis is given by the function x= t^2+6.00t+ 3.00 , where t is in s.
Where is the particle when v(x)= 5.00m/s ?
take the first derivative of f(x) to get velocity.
v=2t+6
Now put v =5, and solve for t.
By the way, v is a function of t, not x.
To find where the particle is located when its velocity is 5.00 m/s, we need to find the value of x when v(x) = 5.00 m/s.
The velocity of a particle is the derivative of its position with respect to time. So, we need to find the derivative of x(t) to get v(t):
v(t) = d(x(t))/dt
To find the derivative of x(t), we can use the power rule of differentiation.
The power rule states that if x = t^n, then the derivative of x with respect to t is dx/dt = n * t^(n-1).
In this case, x(t) = t^2 + 6.00t + 3.00. Taking the derivative of x(t), we get:
dx/dt = d(t^2 + 6.00t + 3.00)/dt
= 2t + 6.00
Now, we have the expression for the velocity of the particle: v(t) = 2t + 6.00.
To find where the particle is when its velocity is 5.00 m/s, we can set v(t) = 5.00 and solve for t:
2t + 6.00 = 5.00
Subtracting 6.00 from both sides:
2t = 5.00 - 6.00
2t = -1.00
Dividing both sides by 2:
t = -1.00 / 2
t = -0.50
Now that we have the value of t, we can substitute it back into the position function x(t) to find the position of the particle:
x(t) = t^2 + 6.00t + 3.00
x(-0.50) = (-0.50)^2 + 6.00(-0.50) + 3.00
= 0.25 - 3.00 + 3.00
= 0.25
Therefore, the particle is located at x = 0.25 meters when its velocity is 5.00 m/s.