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.
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
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1 answer