Math / Algebra

QuestionKinetic energy varies jointly as the mass and the square of the velocity. A mass of 6 grams and velocity of 3 centimeters per second has a kinetic energy of 27 ergs. Find the kinetic energy for a mass of 2 grams and velocity of 6 centimeters per second.
A mass of 2 grams and velocity of 6 centimeters per second has a kinetic energy of $\boxed{}$ ergs.

Studdy Solution

STEP 1

What is this asking? If we know the kinetic energy of a moving object with a certain mass and speed, can we figure out the kinetic energy of another object with a different mass and speed, assuming kinetic energy, mass, and speed are related in a specific way? Watch out! Don't forget to square the velocity!
It's a common mistake to miss that little detail.
Also, "varies jointly" has a specific mathematical meaning, so let's make sure we understand what that means!

STEP 2

1. Define the relationship
2. Find the constant of proportionality
3. Calculate the new kinetic energy

STEP 3

Alright, so "kinetic energy varies jointly as the mass and the square of the velocity." What does that even mean?! It means kinetic energy is directly proportional to both the mass and the square of the velocity.
In simpler terms, if one goes up, the others go up too!

STEP 4

Mathematically, we can write this relationship as: $KE = k \cdot m \cdot v^2$ Where $KE$ is the kinetic energy, $m$ is the mass, $v$ is the velocity, and $k$ is our mysterious constant of proportionality.
We gotta find that guy!

STEP 5

We're given that a mass of 6 grams and a velocity of 3 centimeters per second results in a kinetic energy of 27 ergs.
Let's plug those values into our formula: $27 = k \cdot 6 \cdot 3^2$

STEP 6

Now, let's simplify and solve for $k$ : $27 = k \cdot 6 \cdot 9$ $27 = k \cdot 54$ To isolate $k$ , we divide both sides by 54: $\frac{27}{54} = \frac{k \cdot 54}{54}$ $\frac{1}{2} = k$ So, our constant of proportionality, $k$ , is $\frac{1}{2}$ !

STEP 7

Now that we know $k = \frac{1}{2}$ , we can use our formula to find the kinetic energy for a mass of 2 grams and a velocity of 6 centimeters per second: $KE = \frac{1}{2} \cdot m \cdot v^2$ $KE = \frac{1}{2} \cdot 2 \cdot 6^2$

STEP 8

Let's crunch those numbers: $KE = \frac{1}{2} \cdot 2 \cdot 36$ $KE = 1 \cdot 36$ $KE = 36$ Boom! The kinetic energy is 36 ergs!

STEP 9

The kinetic energy for a mass of 2 grams and velocity of 6 centimeters per second is 36 ergs.

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Studdy Solution

STEP 1

STEP 2

1. Define the relationship
2. Find the constant of proportionality
3. Calculate the new kinetic energy

STEP 3

Alright, so "kinetic energy varies jointly as the mass and the square of the velocity." What does that even mean?! It means kinetic energy is directly proportional to both the mass and the square of the velocity.
In simpler terms, if one goes up, the others go up too!

STEP 4

Mathematically, we can write this relationship as: $KE = k \cdot m \cdot v^2$ Where $KE$ is the kinetic energy, $m$ is the mass, $v$ is the velocity, and $k$ is our mysterious constant of proportionality.
We gotta find that guy!

STEP 5

We're given that a mass of 6 grams and a velocity of 3 centimeters per second results in a kinetic energy of 27 ergs.
Let's plug those values into our formula: $27 = k \cdot 6 \cdot 3^2$

STEP 6

Now, let's simplify and solve for $k$ : $27 = k \cdot 6 \cdot 9$ $27 = k \cdot 54$ To isolate $k$ , we divide both sides by 54: $\frac{27}{54} = \frac{k \cdot 54}{54}$ $\frac{1}{2} = k$ So, our constant of proportionality, $k$ , is $\frac{1}{2}$ !

STEP 7

Now that we know $k = \frac{1}{2}$ , we can use our formula to find the kinetic energy for a mass of 2 grams and a velocity of 6 centimeters per second: $KE = \frac{1}{2} \cdot m \cdot v^2$ $KE = \frac{1}{2} \cdot 2 \cdot 6^2$

STEP 8

Let's crunch those numbers: $KE = \frac{1}{2} \cdot 2 \cdot 36$ $KE = 1 \cdot 36$ $KE = 36$ Boom! The kinetic energy is 36 ergs!

STEP 9

The kinetic energy for a mass of 2 grams and velocity of 6 centimeters per second is 36 ergs.

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Studdy Solution

STEP 1

STEP 2

1. Define the relationship2. Find the constant of proportionality3. Calculate the new kinetic energy

STEP 3

Alright, so "kinetic energy varies jointly as the mass and the square of the velocity." What does *that* even mean?! It means kinetic energy is *directly proportional* to both the mass and the square of the velocity.In simpler terms, if one goes up, the others go up too!

STEP 4

Mathematically, we can write this relationship as: KE=k⋅m⋅v2 KE = k \cdot m \cdot v^2 KE=k⋅m⋅v2 Where KEKEKE is the **kinetic energy**, mmm is the **mass**, vvv is the **velocity**, and kkk is our mysterious **constant of proportionality**.We gotta find *that* guy!

STEP 5

We're given that a mass of **6 grams** and a velocity of **3 centimeters per second** results in a kinetic energy of **27 ergs**.Let's plug those values into our formula: 27=k⋅6⋅32 27 = k \cdot 6 \cdot 3^2 27=k⋅6⋅32

STEP 6

STEP 7

STEP 8

Let's crunch those numbers: KE=12⋅2⋅36 KE = \frac{1}{2} \cdot 2 \cdot 36 KE=21​⋅2⋅36 KE=1⋅36 KE = 1 \cdot 36 KE=1⋅36KE=36 KE = 36 KE=36Boom! The kinetic energy is **36 ergs**!

STEP 9

The kinetic energy for a mass of 2 grams and velocity of 6 centimeters per second is 36 ergs.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Define the relationship
2. Find the constant of proportionality
3. Calculate the new kinetic energy

Alright, so "kinetic energy varies jointly as the mass and the square of the velocity." What does that even mean?! It means kinetic energy is directly proportional to both the mass and the square of the velocity.
In simpler terms, if one goes up, the others go up too!

Mathematically, we can write this relationship as: $KE = k \cdot m \cdot v^2$ Where $KE$ is the kinetic energy, $m$ is the mass, $v$ is the velocity, and $k$ is our mysterious constant of proportionality.
We gotta find that guy!

We're given that a mass of 6 grams and a velocity of 3 centimeters per second results in a kinetic energy of 27 ergs.
Let's plug those values into our formula: $27 = k \cdot 6 \cdot 3^2$

Let's crunch those numbers: $KE = \frac{1}{2} \cdot 2 \cdot 36$ $KE = 1 \cdot 36$ $KE = 36$ Boom! The kinetic energy is 36 ergs!