Given that [tex]m=\frac{1}{s} \sqrt{\frac{g x}{z}}[/tex]. Find [tex]z[/tex] in terms of [tex]m, s, g[/tex] and [tex]x[/tex]
Answer (2)
To isolate z in the equation m = s 1 z gx , we follow several algebraic steps: multiply by s, square both sides, and isolate z to get z = m 2 s 2 gx . This formula shows the relationship between variables in a physics context, especially in dynamics.
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Multiply both sides of the equation by s : m s = z gx .
Square both sides: ( m s ) 2 = z gx , which simplifies to m 2 s 2 = z gx .
Multiply both sides by z : m 2 s 2 z = gx .
Divide both sides by m 2 s 2 to isolate z : z = m 2 s 2 gx .
z = m 2 s 2 gx
Explanation
Understanding the Problem We are given the equation m = s 1 z gx and we want to express z in terms of m , s , g , and x .
Isolating the Square Root First, let's multiply both sides of the equation by s to get rid of the fraction on the right side: m s = z gx
Eliminating the Square Root Next, we square both sides of the equation to eliminate the square root: ( m s ) 2 = z gx m 2 s 2 = z gx
Isolating z Now, we want to isolate z . Multiply both sides of the equation by z :
m 2 s 2 z = gx
Solving for z Finally, divide both sides by m 2 s 2 to solve for z :
z = m 2 s 2 gx
Final Answer Therefore, z in terms of m , s , g , and x is given by: z = m 2 s 2 gx
Examples
In physics, this formula could relate the mass ( m ) of an object to its acceleration ( g ), displacement ( x ), and a medium property ( z ) under certain conditions, with s being a scaling factor. For example, if you know the mass of a projectile, its acceleration due to gravity, and the distance it travels, you can use this formula to determine a property of the medium it's traveling through, such as its density or resistance. Rearranging the formula allows you to solve for this unknown property ( z ) based on the other known quantities. This kind of problem-solving is crucial in fields like aerodynamics and ballistics.
