The kinetic energy of the car of a rollercoaster is given by the formula [tex]$k =\frac{1}{2} m v ^{ 2 }$[/tex], where k is kinetic energy, [tex]$m$[/tex] is the mass of the car, and [tex]$v$[/tex] is the velocity of the car. Solving this formula for [tex]$v$[/tex], we get [tex]$v=\sqrt{\frac{2 k}{m}}$[/tex]. Which formula gives the velocity of the car in simplest form?
[tex]$v=\frac{\sqrt{2 k m}}{2 m}$[/tex]
[tex]$v=\frac{\sqrt{2 k m}}{m}$[/tex]
[tex]$v=\frac{\sqrt{2 k}}{m}$[/tex]

The velocity of the rollercoaster car can be expressed as v = m 2 km ​ ​ , which matches the second option given. This formula is derived from the kinetic energy formula by rearranging terms and simplifying. Therefore, the simplest form of the velocity formula is v = m 2 km ​ ​ .
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Answered by Anonymous | 2025-07-04

Start with the given formula: v = m 2 k ​ ​ .
Multiply the fraction inside the square root by m m ​ : v = m 2 k ​ ⋅ m m ​ ​ = m 2 2 km ​ ​ .
Simplify the expression: v = m 2 ​ 2 km ​ ​ = m 2 km ​ ​ .
The simplest form of the velocity formula is: v = m 2 km ​ ​ ​ .

Explanation

Understanding the Problem We are given the formula for the velocity of a rollercoaster car: v = m 2 k ​ ​ , where k is the kinetic energy and m is the mass of the car. Our goal is to simplify this expression by rationalizing the denominator.

Rationalizing the Denominator To rationalize the denominator, we want to eliminate the square root from the denominator. We can do this by multiplying the fraction inside the square root by m m ​ :


v = m 2 k ​ ⋅ m m ​ ​ = m 2 2 km ​ ​

Simplifying the Expression Now we can simplify the expression by taking the square root of the denominator:

v = m 2 ​ 2 km ​ ​ = m 2 km ​ ​

Final Answer Comparing our simplified expression, v = m 2 km ​ ​ , with the given options, we see that it matches the second option.

Therefore, the simplest form of the velocity formula is:
v = m 2 km ​ ​
Examples
Understanding how to simplify radical expressions like the velocity formula v = m 2 k ​ ​ is useful in various physics and engineering applications. For instance, when designing a rollercoaster, engineers need to calculate the velocity of the cars at different points. Simplifying the formula makes it easier to compute and analyze the velocity, ensuring the rollercoaster is safe and exciting. This skill is also applicable in other areas, such as calculating the speed of projectiles or analyzing the motion of objects in various mechanical systems.

Answered by GinnyAnswer | 2025-07-04