Answer :
let's assume that m1 is the mass of the first particle and m2 is the mass of the second particle, with m1 not equal to m2.
The kinetic energy (KE) of an object can be calculated using the formula KE = 0.5 * mass * velocity^2.
Now, if both particles have the same velocity, the one with the greater mass will have the greater kinetic energy. This can be proven mathematically:
Let's say v is the velocity of both particles.
For m1:
KE1 = 0.5 * m1 * v^2
For m2:
KE2 = 0.5 * m2 * v^2
Since m1 is not equal to m2, let's assume without loss of generality that m1 > m2.
Now, let's compare KE1 and KE2:
KE1 = 0.5 * m1 * v^2 > 0.5 * m2 * v^2 = KE2
Therefore, the particle with mass m1 will have the greater kinetic energy when both particles have the same velocity.
Answer:
Refer below.
Explanation:
Given that we have two masses with a mass 'm₁' and 'm₂' where m₁ ≠ m₂, show that one has a greater kinetic energy than the other.
To determine which particle has the greater kinetic energy, we need to compare their kinetic energies given by the formula:
[tex]\boxed{ \begin{array}{ccc} \text{\underline{Kinetic Energy:}} \\\\ K = \dfrac{1}{2}mv^2 \\\\ \text{Where:} \\ \bullet \ K \ \text{is the kinetic energy} \\ \bullet \ m \ \text{is the mass of the object} \\ \bullet \ v \ \text{is the velocity of the object} \end{array}}[/tex]
Case 1: Equal Velocities
If the two particles, 'm₁' and 'm₂', have the same velocity 'v':
[tex]K_1=\dfrac{1}{2}m_1v^2[/tex]
[tex]K_2=\dfrac{1}{2}m_2v^2[/tex]
To compare the kinetic energies:
[tex]\Longrightarrow K_1-K_2=\dfrac{1}{2}m_1v^2-\dfrac{1}{2}m_1v^2\\\\\\\\\Longrightarrow K_1-K_2=\dfrac{1}{2}v^2(m_1-m_2)[/tex]
Since v² and 1/2 are positive constants:
- If m₁ > m₂, then K₁ > K₂
- If m₁ < m₂, then K₁ < K₂
Thus, the particle with the greater mass has the greater kinetic energy when the velocities are equal.
Case 2: Different Velocities
If the particles have different velocities 'v₁' and 'v₂':
[tex]K_1=\dfrac{1}{2}m_1v_1^2[/tex]
[tex]K_2=\dfrac{1}{2}m_2v_2^2[/tex]
To determine which particle has the greater kinetic energy, we need to compare 1/2m₁v₁² and 1/2m₂v₂². There isn't exactly a straightforward comparison without specific values for m₁, m₂, v₁, and v₂. Generally, the particle with the larger product 'm' and 'v²' will have the greater kinetic energy.
Lets have:
- m₁ = 2 kg, v₁ = 3 m/s
- m₂ = 1 kg, v₂ = 4 m/s
Then...
[tex]K_1=\dfrac{1}{2}(2)(3)^2 = \boxed{9 \text{ J}} \text{ and } K_2=\dfrac{1}{2}(1)(4)^2=\boxed{8 \text{ J}}[/tex]
But if we have:
- m₁ = 2 kg, v₁ = 4 m/s
- m₂ = 1 kg, v₂ = 3 m/s
[tex]K_1=\dfrac{1}{2}(2)(4)^2 = \boxed{16 \text{ J}} \text{ and } K_2=\dfrac{1}{2}(1)(3)^2=\boxed{4.5 \text{ J}}[/tex]
Thus, the greater kinetic energy depends on the specific values of the masses and velocities of the particles.
