The position of two particles p1 and the end of t seconds is given by s1=3t^3-12t^2+18t+5, s2=-t^4+9t^2-12t. When do the two particles have the same velocity
To find when the two particles have the same velocity, we need to find the time at which the derivatives of their position functions are equal.
Let's start by finding the derivative of s1:
s1' = d/dt (3t^3-12t^2+18t+5)
= 9t^2 - 24t + 18
Next, let's find the derivative of s2:
s2' = d/dt (-t^4+9t^2-12t)
= -4t^3 + 18t - 12
To find when the two particles have the same velocity, we set s1' equal to s2' and solve for t:
9t^2 - 24t + 18 = -4t^3 + 18t - 12
Rearranging the equation, we get:
4t^3 + 9t^2 + 6t - 30 = 0
This is a cubic equation that needs to be solved to find the values of t when the two particles have the same velocity.
Unfortunately, solving cubic equations can be quite involved and doesn't have a simple solution method like quadratics. One option is to use numerical methods, such as using a graphing calculator or computer software to find the approximate values of t that satisfy the equation.
Alternatively, if an exact solution is required, we can use the Rational Root Theorem to find any rational roots of the equation, and then apply polynomial division to simplify it into a quadratic equation, which can be solved using the quadratic formula.
Overall, the exact values of t when the two particles have the same velocity will depend on the specific roots of the cubic equation 4t^3 + 9t^2 + 6t - 30 = 0.
To find when the two particles have the same velocity, we need to find when their velocities are equal. Velocity is the derivative of position with respect to time.
Let's start by finding the velocities of the two particles.
The velocity of particle p1 is given by taking the derivative of its position function s1:
v1 = d(s1)/dt = d/dt (3t^3 - 12t^2 + 18t + 5)
Differentiating each term, we get:
v1 = 9t^2 - 24t + 18
Now let's find the velocity of particle p2. Similarly, we take the derivative of its position function s2:
v2 = d(s2)/dt = d/dt(-t^4 + 9t^2 - 12t)
Again, differentiating each term, we get:
v2 = -4t^3 + 18t - 12
To find when the two particles have the same velocity, we set v1 equal to v2 and solve for t:
9t^2 - 24t + 18 = -4t^3 + 18t - 12
Simplifying and rearranging the equation:
4t^3 + 9t^2 - 42t + 30 = 0
Unfortunately, this is a cubic equation, and solving it exactly can be quite complex. However, we can use numerical methods or approximations to find an approximate solution.
One numerical method is to use a graphing calculator or software to plot the two functions v1 and v2 and find their intersection point. Another method is to use approximation techniques such as Newton's method or the method of bisection to find an approximate solution.
Alternatively, if there are answer choices given, you can substitute each answer choice back into the equation and check if it satisfies the equation.
Please let me know if you would like to proceed with any of these methods or if you have any further questions.