How Do You Calculate Time When Velocity Equals Zero in a Complex Equation?

[math4life]

Homework Statement

This is more of a math question - I just want to know how I can find the value of t when v=0.

0=5(t^3/2)-(27/2)(t^1/2)+9(t^-1/2)

Homework Equations

The Attempt at a Solution

I do not know how to solve this. Please show steps. Thanks.

 

[berkeman]

Hint: What happens if you multiply through by $$\sqrt{t}$$ ?

Homework Statement

This is more of a math question - I just want to know how I can find the value of t when v=0.

0=5(t^3/2)-(27/2)(t^1/2)+9(t^-1/2)

Homework Equations

The Attempt at a Solution

I do not know how to solve this. Please show steps. Thanks.

 

[nlightner]

I would approach this problem by first identifying the variables and their units. In this case, the variables are time (t) and velocity (v). The units of time are typically measured in seconds (s) and the units of velocity are typically measured in meters per second (m/s).

Next, I would look at the given equation and try to simplify it as much as possible. In this case, it may be helpful to rewrite the equation using exponents instead of fractional powers:

0 = 5t^(3/2) - (27/2)t^(1/2) + 9t^(-1/2)

0 = 5t^(3/2) - (27/2)t^(1/2) + 9/t^(1/2)

Now, we can see that the equation has a quadratic form, with t^(1/2) as the variable. We can use the quadratic formula to solve for t^(1/2):

t^(1/2) = (-b ± √(b^2-4ac)) / 2a

Where a = 5, b = -(27/2), and c = 9.

Plugging in these values, we get:

t^(1/2) = (-(-(27/2)) ± √((-(27/2))^2 - 4(5)(9))) / 2(5)

t^(1/2) = (27/2 ± √(729/4 - 180)) / 10

t^(1/2) = (27/2 ± √(549/4)) / 10

Now, we can solve for t by squaring both sides:

t = ((27/2) ± √(549/4)) / 10)^2

t = (729/4 ± 549/4) / 100

t = (1278/4) / 100 or t = (180/4) / 100

t = 319.5/100 or t = 45/100

Therefore, the possible values of t are t = 3.195 seconds or t = 0.45 seconds. However, we must remember that t^(1/2) was just a variable in our simplified equation. To find the actual values of t, we need to substitute these values back into the original equation:

v = 5t