Find the ratio of the linear momenta of two particles of masses 1 kg and 4 kg if their kinetic energies are equal.

To find the ratio of the linear momenta of two particles with different masses but equal kinetic energies, we can start by recalling the formulas for kinetic energy and momentum. The kinetic energy (KE) of an object is given by the equation:

Kinetic Energy Formula

KE = (1/2)mv²

where m is the mass and v is the velocity of the object.

Momentum Formula

The momentum (p) of an object is defined as:

p = mv

Setting Up the Problem

Let’s denote the two particles as Particle A (mass = 1 kg) and Particle B (mass = 4 kg). Since their kinetic energies are equal, we can set up the following equation:

KE_A = KE_B

(1/2)(1 kg)(v_A)² = (1/2)(4 kg)(v_B)²

Simplifying the Equation

We can simplify this equation by canceling out the (1/2) factor:

(1 kg)(v_A)² = (4 kg)(v_B)²

Expressing Velocities

Now, we can express the velocities in terms of each other:

v_A² = 4(v_B)²

Taking the square root of both sides gives us:

v_A = 2v_B

Calculating Momentum

Now that we have a relationship between the velocities, we can find the momenta of both particles:

For Particle A:

p_A = m_Av_A = (1 kg)(2v_B) = 2 kg·v_B

For Particle B:

p_B = m_Bv_B = (4 kg)(v_B) = 4 kg·v_B

Finding the Ratio of Momenta

Now we can find the ratio of the momenta:

Ratio = p_A / p_B = (2 kg·v_B) / (4 kg·v_B)

Notice that the v_B cancels out:

Ratio = 2 kg / 4 kg = 1/2

Final Result

The ratio of the linear momenta of the two particles is 1:2. This means that for every unit of momentum that Particle A has, Particle B has two units of momentum, reflecting the differences in their masses and the fact that they have equal kinetic energies.