Two cars are moving. The first car has twice the mass of the second car but only half as much kinetic energy. When both cars increase their speed by 4.54 m/s, they then have the same kinetic energy. Calculate the original speeds of the two cars.

first car 1
second car 2

Let's assume the initial speed of the first car is v1 and the initial speed of the second car is v2. We can use the formula for kinetic energy:

KE = (1/2) * m * v^2

Given that the first car has twice the mass of the second car and only half the kinetic energy, we can set up the following equations:

(1/2) * (2m) * v1^2 = (1/2) * m * v2^2
(1/2) * (2m) * (v1 + 4.54)^2 = (1/2) * m * (v2 + 4.54)^2

Simplifying the equations:

2v1^2 = v2^2
2(v1 + 4.54)^2 = (v2 + 4.54)^2

Expanding (v1 + 4.54)^2 and (v2 + 4.54)^2:

2v1^2 = v2^2
2v1^2 + 18.16v1 + 20.656 = v2^2 + 9.08v2 + 18.16

Now, let's solve these equations step-by-step.

1. Subtract 2v1^2 and v2^2 from both sides of the first equation:

0 = v2^2 - 2v1^2

2. Simplify the second equation:

2v1^2 + 18.16v1 + 20.656 - v2^2 - 9.08v2 - 18.16 = 0

3. Substitute v2^2 - 2v1^2 into the second equation:

2v1^2 + 18.16v1 + 20.656 - (v2^2 - 2v1^2) - 9.08v2 - 18.16 = 0

4. Combine like terms:

0.16v2^2 + 27.24v1 + 20.656 - 9.08v2 - 18.16 = 0

0.16v2^2 - 9.08v2 + 27.24v1 + 2.496 = 0

5. Now we have a quadratic equation in terms of v2. Let's solve it:

Using the quadratic formula:
v2 = (-(-9.08) ± √((-9.08)^2 - 4 * 0.16 * (27.24v1 + 2.496))) / (2 * 0.16)

Simplifying the equation further:

v2 = (9.08 ± √(82.7264 - 4 * 0.16 * (27.24v1 + 2.496))) / 0.32

v2 = (9.08 ± √(82.7264 - 1.728 * (27.24v1 + 2.496))) / 0.32

v2 = (9.08 ± √(82.7264 - 46.9376v1 - 7.276608)) / 0.32

v2 = (9.08 ± √(35.7888 - 46.9376v1)) / 0.32

6. Now, let's solve for v1 using the first equation:

0 = v2^2 - 2v1^2

0 = (v1 + 4.54)^2 - 2v1^2

Expanding the equation further:

0 = v1^2 + 9.08v1 + 20.6616 - 2v1^2

- v1^2 + 9.08v1 + 20.6616 = 0

7. Now we have another quadratic equation in terms of v1. Let's solve it using the quadratic formula:

v1 = (-9.08 ± √(9.08^2 - 4 * (-1) * 20.6616)) / (-2)

v1 = (-9.08 ± √(82.7264 + 82.6464)) / (-2)

v1 = (-9.08 ± √(165.3728)) / (-2)

v1 = (-9.08 ± 12.855) / (-2)

v1 = (-9.08 + 12.855) / (-2) or v1 = (-9.08 - 12.855) / (-2)

v1 = 1.7695 or v1 = -11.9625

Since speed cannot be negative, we discard v1=-11.9625.

Therefore, the original speed of the first car (v1) is 1.7695 m/s.

Using this value for v1, let's calculate v2:

v2 = (9.08 ± √(35.7888 - 46.9376 * 1.7695)) / 0.32

v2 = (9.08 ± √(35.7888 - 82.91971072)) / 0.32

v2 = (9.08 ± √(-47.13091072)) / 0.32

Since the value inside the square root is negative, it means there is no real solution for v2 using this value for v1.

So, there is no valid solution using the given information.

To solve this problem, we'll need to use the equation for kinetic energy:

Kinetic Energy = 1/2 * mass * velocity^2

Let's start by assigning variables to the given information:

Let the mass of the first car be m1.
Let the mass of the second car be m2.
Let the original velocity of the first car be v1.
Let the original velocity of the second car be v2.

According to the problem, the mass of the first car (m1) is twice the mass of the second car (m2). So we can write this as:

m1 = 2m2 ...(Equation 1)

The problem also states that the first car has half the kinetic energy of the second car. Mathematically, this can be expressed as:

1/2 * m1 * v1^2 = 2 * (1/2 * m2 * v2^2)
Simplifying this equation, we get:

1/2 * m1 * v1^2 = m2 * v2^2
Rearranging this equation, we have:

v1^2 = 2v2^2 ...(Equation 2)

The problem further mentions that when both cars increase their speed by 4.54 m/s, they then have the same kinetic energy. Mathematically, this can be expressed as:

1/2 * m1 * (v1 + 4.54)^2 = 1/2 * m2 * (v2 + 4.54)^2

Since we have two equations (Equation 1 and Equation 2) with two unknowns (v1 and v2), we can solve them simultaneously.

Substituting Equation 1 into Equation 2, we get:

(2v2^2) = 2v2^2
This equation shows us that the velocities are equal.

From Equation 1, we know that m1 = 2m2.

Therefore, we can write:

(m2*(v2 + 4.54)^2) = (2m2*(v2)^2)

Canceling out m2, we get:

(v2 + 4.54)^2 = 2 * (v2)^2

Expanding and simplifying, we have:

v2^2 + 2 * 4.54 * v2 + (4.54)^2 = 2v2^2

Rearranging this equation, we obtain:

v2^2 - 2 * 4.54 * v2 - (4.54)^2 = 0

Now we can solve this quadratic equation to find the value of v2.

Using the quadratic formula:

v2 = [-(-2 * 4.54) ± √((-2 * 4.54)^2 - 4 * 1 * (-4.54)^2)] / (2 * 1)

Simplifying this equation, we get:

v2 = [9.08 ± √(9.08^2 - 4 * (-4.54)^2)] / 2

v2 = [9.08 ± √(82.7264 + 82.7264)] / 2

v2 = [9.08 ± √(165.4528)] / 2

v2 = [9.08 ± 12.8552] / 2

The two potential values for v2 are:

v2 = (9.08 + 12.8552) / 2 = 10.9676 m/s
v2 = (9.08 - 12.8552) / 2 = -1.3876 m/s (Discard this negative value)

So, the original speed of the second car (v2) is approximately 10.9676 m/s.

Now, to find v1, we can use Equation 1. Substituting the value of v2, we get:

m1 = 2m2
v1^2 = 2v2^2

Therefore:

v1^2 = 2 * (10.9676)^2

v1^2 = 2 * 120.1409

v1^2 = 240.2818

Taking the square root of both sides, we obtain:

v1 = √(240.2818)

v1 is approximately 15.49 m/s.

So, the original speeds of the two cars are approximately 15.49 m/s (for the first car) and 10.9676 m/s (for the second car).