A 2.5-kg ball and a 5.0-kg ball have an elastic collision. Before the collision, the 2.5-kg ball was at rest and the other ball had a speed of 3.5 m/s. What is the kinetic energy of the 2.5-kg ball after the collision?

Options

A. 1.7 J
B. 3.4 J
C. 8.1 J
D. 27 J

Answer

To solve the problem of the elastic collision between a 2.5-kg ball and a 5.0-kg ball, we will follow these steps:

### Step 1: Understand the Problem
We have two balls:
- Ball 1 (m1 = 2.5 kg) is at rest (u1 = 0 m/s).
- Ball 2 (m2 = 5.0 kg) is moving with a speed of 3.5 m/s (u2 = 3.5 m/s).

We need to find the kinetic energy of Ball 1 after the collision.

### Step 2: Use Conservation of Momentum
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision.

The momentum before the collision:
\[ p_{initial} = m_1 \cdot u_1 + m_2 \cdot u_2 = 2.5 \cdot 0 + 5.0 \cdot 3.5 = 17.5 \, \text{kg m/s} \]

Let \( v_1 \) be the final velocity of Ball 1 and \( v_2 \) be the final velocity of Ball 2 after the collision. The momentum after the collision is:
\[ p_{final} = m_1 \cdot v_1 + m_2 \cdot v_2 \]

Setting the initial and final momentum equal:
\[ 17.5 = 2.5 \cdot v_1 + 5.0 \cdot v_2 \quad \text{(1)} \]

### Step 3: Use Conservation of Kinetic Energy
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.

The kinetic energy before the collision:
\[ KE_{initial} = \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} \cdot 2.5 \cdot 0^2 + \frac{1}{2} \cdot 5.0 \cdot (3.5)^2 \]
\[ = 0 + \frac{1}{2} \cdot 5.0 \cdot 12.25 = 30.625 \, \text{J} \]

The kinetic energy after the collision:
\[ KE_{final} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 \quad \text{(2)} \]

Setting the initial and final kinetic energy equal:
\[ 30.625 = \frac{1}{2} \cdot 2.5 \cdot v_1^2 + \frac{1}{2} \cdot 5.0 \cdot v_2^2 \quad \text{(3)} \]

### Step 4: Solve the Equations
From equation (1):
\[ 5.0 \cdot v_2 = 17.5 - 2.5 \cdot v_1 \]
\[ v_2 = \frac{17.5 - 2.5 \cdot v_1}{5.0} \quad \text{(4)} \]

Substituting equation (4) into equation (3):
\[ 30.625 = \frac{1}{2} \cdot 2.5 \cdot v_1^2 + \frac{1}{2} \cdot 5.0 \cdot \left(\frac{17.5 - 2.5 \cdot v_1}{5.0}\right)^2 \]

This will lead to a quadratic equation in terms of \( v_1 \). After solving the quadratic equation, we can find the value of \( v_1 \).

### Step 5: Calculate Kinetic Energy of Ball 1
Once we find \( v_1 \), we can calculate the kinetic energy of the 2.5-kg ball after the collision using:
\[ KE_{final} = \frac{1}{2} m_1 v_1^2 \]