One car has twice the mass of a second car, but only half as much kinetic energy. When both cars increase their speed by 5.0m/s, they then have the same kinetic energy.

What were the original speeds of the two cars?

Let's assume the mass of the second car is "m" and its original speed is "v".

According to the problem, the first car has twice the mass of the second car, which means its mass is 2m.

The kinetic energy (KE) is given by the formula KE = (1/2) * mass * velocity^2.

For the second car, its initial kinetic energy is KE2 = (1/2) * m * v^2.

Now, let's calculate the initial kinetic energy (KE1) of the first car:
KE1 = (1/2) * (2m) * v^2 = m * v^2.

We are also given that the first car has only half the kinetic energy of the second car:
m * v^2 = (1/2) * m * (5+v)^2.

Expanding the equation:
m * v^2 = (1/2) * m * (5+v)^2
2 * v^2 = (5+v)^2
4 * v^2 = 25 + 10v + v^2
3 * v^2 - 10v - 25 = 0.

Now, using the quadratic formula to solve for v:
v = (-b ± √(b^2 - 4ac)) / (2a).

Plugging in a = 3, b = -10, and c = -25:
v = (-(-10) ± √((-10)^2 - 4(3)(-25))) / (2(3))
v = (10 ± √(100 + 300)) / 6
v = (10 ± √400) / 6
v = (10 ± 20) / 6.

Simplifying:
v = (10 + 20) / 6 = 30 / 6 = 5, or
v = (10 - 20) / 6 = -10 / 6 = -5/3.

Since speed cannot be negative, the initial speed of the second car is v = 5 m/s. Therefore, the initial speed of the first car is twice that, which is 2 * 5 = 10 m/s.

Thus, the original speeds of the two cars are 10 m/s and 5 m/s.

To solve this problem, we can use the formulas for kinetic energy and the relationship between kinetic energy and speed.

Let's assume the mass of the second car is "m" and its original speed is "v". So, the mass of the first car would be "2m" and its original speed would be "u" (unknown).

First, let's calculate the initial kinetic energy of each car:
- The initial kinetic energy of the first car (KE₁) would be: KE₁ = (1/2)(mass₁)(velocity₁)² = (1/2)(2m)(u²) = mu².
- The initial kinetic energy of the second car (KE₂) would be: KE₂ = (1/2)(mass₂)(velocity₂)² = (1/2)(m)(v²) = mv².

Given that the first car has only half the kinetic energy of the second car, we have the equation: mu² = (1/2)mv².

Next, let's calculate the final kinetic energy of each car after both increase their speed by 5.0m/s:
- The final kinetic energy of the first car (KE₁') would be: KE₁' = (1/2)(mass₁)(velocity₁')² = (1/2)(2m)((u + 5.0)²) = m(u + 5.0)².
- The final kinetic energy of the second car (KE₂') would be: KE₂' = (1/2)(mass₂)(velocity₂')² = (1/2)(m)((v + 5.0)²) = m(v + 5.0)².

Given that both cars have the same kinetic energy after the increase, we have the equation: m(u + 5.0)² = m(v + 5.0)².

Let's simplify the equation and solve for "u":
(u + 5.0)² = (v + 5.0)²
u + 5.0 = v + 5.0 (taking the square root on both sides)
u = v

Since we know that the cars have different masses, but the same speed, we can conclude that the original speeds of the two cars were equal.

Therefore, the original speeds of the two cars were the same but unknown.

M1 is the heavier car and the mass of the other car is M2 = M1/2

Orginally,
(1/2) M1 V1^2 = (1/2)*(1/2)(M1/2)V2^2

which implies that
V1^2 = (1/4)V2^2
V1 = V2/2

Also,
(1/2)M1*(V1+5)^2 = (1/2)(M1/2)(V2+5)^2
= (1/2)(M1/2)(2V1 + 5)^2
(V1+5)^2 = (1/2)(2V1+5)^2
V1^2 + 10 V1 + 25 = 2V1^2 + 10 V1 + 25/2
V1^2 = 25/2
V1 = 5/sqrt2
V2 = 5 sqrt2