a particle moves along the x-axis for t>0 with an acceleration of a(t)=12t+6 where t is time in seconds. the particle's velocity at t=36 cm/sec. The initial position of the particle is 4 cm. What is the position of the particle when the velocity is zero
To find the position of the particle when the velocity is zero, we need to first find the time when the velocity becomes zero and then plug that time into the position equation.
Given that the velocity at t=36 is 36 cm/s, we can set up the equation:
v(t) = 36 = dx(t)/dt
where v(t) represents the velocity and dx(t)/dt represents the derivative of the position function x(t) with respect to time.
Now, let's integrate the acceleration function a(t) to find the velocity function v(t):
a(t) = 12t + 6
Integrating a(t) will give us:
v(t) = ∫(12t + 6) dt = 6t^2 + 6t + C
where C is the integration constant.
Since the velocity at t=36 is 36 cm/s, we can now solve for C using this information:
v(36) = 6(36)^2 + 6(36) + C = 36
C = 0 - 6(36)^2 - 6(36) = -7776
So, the velocity function becomes:
v(t) = 6t^2 + 6t - 7776
Now, we need to find the time when the velocity is zero. We can do this by setting v(t) = 0 and solving for t:
0 = 6t^2 + 6t - 7776
This equation is a quadratic equation. We can solve it by factoring or by using the quadratic formula.
Using the quadratic formula, we have:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values of a, b, and c, we got:
t = (-6 ± √(6^2 - 4(6)(-7776))) / (2(6))
Simplifying further, we have:
t = (-6 ± √(36 + 186624)) / 12
t = (-6 ± √186660) / 12
t = (-6 ± 432) / 12
Solving for t, we get two possible solutions:
t1 = (-6 + 432) / 12 = 426 / 12 = 35.5 seconds
t2 = (-6 - 432) / 12 = -438 / 12 = -36.5 seconds (not valid, as time cannot be negative)
Therefore, the time when the velocity is zero is t = 35.5 seconds.
Finally, we can plug this value of t into the position function to find the position of the particle:
x(t) = ∫(v(t)) dt = ∫((6t^2 + 6t - 7776)) dt
Integrating x(t) will give us:
x(t) = 2t^3 + 3t^2 - 7776t + C
where C is the integration constant.
To find C, we can use the initial position of the particle, which is given as x(0) = 4 cm:
x(0) = 2(0)^3 + 3(0)^2 - 7776(0) + C
C = 4
So, the position function becomes:
x(t) = 2t^3 + 3t^2 - 7776t + 4
Now, plugging in t = 35.5 seconds, we can find the position of the particle when the velocity is zero:
x(35.5) = 2(35.5)^3 + 3(35.5)^2 - 7776(35.5) + 4
x(35.5) = 14122.25 + 3989.25 - 275736 + 4
x(35.5) ≈ -234620.5 cm
Therefore, the position of the particle when the velocity is zero is approximately -234620.5 cm.