What is the speed ratio (vA/vB) if particle A has 6 times the kinetic energy and half the mass of particle B?
3.46
1.73
2.45
0.289
Well, based on particle A having 6 times the kinetic energy and half the mass of particle B, we can use the equation for kinetic energy: KE = (1/2)mv^2.
Since particle A has 6 times the kinetic energy, we can write it as:
(1/2)(ma)(vA^2) = 6[(1/2)(mb)(vB^2)]
Now, we divide both sides of the equation by (1/2) and (mb) to simplify it:
ma(vA^2) = 12(vB^2)
Then we divide both sides by ma:
vA^2 = 12(vB^2)/ma
And finally, take the square root of both sides:
vA = √(12(vB^2)/ma)
Now, since we're looking for the speed ratio (vA/vB), we can write it as:
vA/vB = (√(12(vB^2)/ma))/vB
Which simplifies to:
vA/vB = √(12/ma)
With the information given, ma = (1/2)mb. Plugging that in:
vA/vB = √(12/((1/2)mb))
Since ma = (1/2)mb, the ratio simplifies to:
vA/vB = √(12/(1/2))
vA/vB = √(12/0.5)
Hold on to your funny bone, because here comes the punchline. Are you ready?
The answer is 3.46...but remember, I'm a bot, not Einstein!
To determine the speed ratio (vA/vB), we need to use the equations for kinetic energy and the relationship between kinetic energy and mass.
The kinetic energy of a particle is given by the equation:
KE = (1/2) * m * v^2
Given that particle A has 6 times the kinetic energy and half the mass of particle B, we can write the following equations:
(1/2) * mA * vA^2 = 6 * ((1/2) * mB * vB^2)
mA * vA^2 = 6 * mB * vB^2
Since mA = (1/2) * mB, we can substitute this into the equation:
(1/2) * mB * vA^2 = 6 * mB * vB^2
Dividing both sides of the equation by mB, we get:
(1/2) * vA^2 = 6 * vB^2
Now we can calculate the speed ratio:
(vA/vB)^2 = (1/12)
vA/vB = √(1/12)
vA/vB ≈ 0.289
Therefore, the speed ratio (vA/vB) is approximately 0.289.
To determine the speed ratio (vA/vB) between particle A and particle B, we need to consider the relationship between kinetic energy, mass, and velocity.
The kinetic energy of an object can be calculated using the formula: KE = (1/2)mv^2, where KE is the kinetic energy, m is the mass, and v is the velocity.
Given that particle A has 6 times the kinetic energy and half the mass of particle B, we can write the following equation:
(1/2)mA(vA)^2 = 6 * (1/2)mB(vB)^2
Since the masses are given in the ratio mA/mB = 1/2, we can substitute mA = (1/2)mB into the equation:
(1/2)(1/2)mB(vA)^2 = 6 * (1/2)mB(vB)^2
Now we can cancel out the terms (1/2)mB on both sides of the equation:
(vA)^2 = 6(vB)^2
Taking the square root of both sides:
vA = sqrt(6) * vB
Therefore, the speed ratio (vA/vB) is sqrt(6), which is approximately 2.45.
Hence, the correct answer is 2.45.
KE = 1/2 m vB^2
6 KE = 1/2 * m/2 * vA^2
12 = (vA / vB)^2