A bullet of mass 10g travelling horizontally at 200 m/s strikes and embeds in a pendulum bob of mass 2.0kg.

Assuming Specific heat capacity at constant volume (Cv) for the bob plus bullet is 3R, calculate the temperature increase of the system due to the collision. Take molecular mass of the system to be 200g/mol.

To find the temperature increase of the system after the bullet embeds itself in the pendulum bob, we can use the principles of conservation of momentum and the relationship between kinetic energy and temperature change. Let's break this down step by step.

Step 1: Calculate the Initial Momentum

The initial momentum of the system can be calculated using the formula:

Momentum (p) = mass (m) × velocity (v)

For the bullet:

• Mass of the bullet, m_bullet = 10 g = 0.01 kg
• Velocity of the bullet, v_bullet = 200 m/s

So, the initial momentum of the bullet is:

p_bullet = 0.01 kg × 200 m/s = 2 kg·m/s

The pendulum bob is initially at rest, so its momentum is:

p_bob = 0 kg·m/s

Step 2: Total Initial Momentum

The total initial momentum of the system (bullet + bob) is simply the momentum of the bullet since the bob is at rest:

p_total = p_bullet + p_bob = 2 kg·m/s + 0 = 2 kg·m/s

Step 3: Calculate the Final Velocity

After the collision, the bullet and bob move together as a single object. We can use conservation of momentum to find the final velocity (v_f):

m_bullet × v_bullet + m_bob × v_bob = (m_bullet + m_bob) × v_f

Substituting the values:

• m_bob = 2.0 kg
• v_bob = 0 m/s

2 kg·m/s = (0.01 kg + 2.0 kg) × v_f

2 kg·m/s = 2.01 kg × v_f

Solving for v_f gives:

v_f = 2 kg·m/s / 2.01 kg ≈ 0.995 m/s

Step 4: Calculate the Kinetic Energy Before and After the Collision

Next, we calculate the kinetic energy before and after the collision:

Kinetic Energy (KE) = 0.5 × m × v²

Initial kinetic energy (KE_initial):

KE_initial = 0.5 × 0.01 kg × (200 m/s)² = 0.5 × 0.01 × 40000 = 200 J

Final kinetic energy (KE_final):

KE_final = 0.5 × 2.01 kg × (0.995 m/s)² ≈ 0.5 × 2.01 × 0.990 = 0.995 J

Step 5: Calculate the Change in Kinetic Energy

The change in kinetic energy (ΔKE) is:

ΔKE = KE_initial - KE_final = 200 J - 0.995 J ≈ 199.005 J

Step 6: Relate the Change in Kinetic Energy to Temperature Increase

The energy lost in the collision is converted into internal energy, which raises the temperature of the system. The specific heat capacity at constant volume (C_v) is given as 3R, where R is the universal gas constant (approximately 8.314 J/(mol·K)).

First, we need to find the number of moles (n) in the system:

n = mass (g) / molecular mass (g/mol)

For the system:

mass = 10 g + 2000 g = 2010 g

n = 2010 g / 200 g/mol = 10.05 mol

Now, we can use the formula for temperature change (ΔT):

ΔT = ΔKE / (n × C_v)

Substituting the values:

ΔT = 199.005 J / (10.05 mol × 3 × 8.314 J/(mol·K))

ΔT = 199.005 J / (10.05 × 24.942) ≈ 199.005 J / 250.0 J/K ≈ 0.796 K

Final Result

The temperature increase of the system due to the collision is approximately 0.796 K. This means that the energy from the bullet's kinetic energy has been transformed into thermal energy, resulting in a slight rise in temperature of the combined system of the bullet and the pendulum bob.