An object of mass 5kg moving with a velocity of 10m/s collides with a 25kg moving with a velocity of 30m/s, calculate the loss in kinetic energy if they coalesce on impact while traveling in the same direction

The initial momentum of the system is the sum of the individual momenta of the two objects.

$$P_{\text{initial}} = m_1 u_1 + m_2 u_2$$

Given:
$m_1 = 5 \text{ kg}$
$u_1 = 10 \text{ m/s}$
$m_2 = 25 \text{ kg}$
$u_2 = 30 \text{ m/s}$

Substitute the values:

$$P_{\text{initial}} = (5 \text{ kg})(10 \text{ m/s}) + (25 \text{ kg})(30 \text{ m/s})$$ $$P_{\text{initial}} = 50 \text{ kg} \cdot \text{m/s} + 750 \text{ kg} \cdot \text{m/s}$$ $$P_{\text{initial}} = 800 \text{ kg} \cdot \text{m/s}$$

Since the objects coalesce, they move together as a single mass ($m_1 + m_2$) with a common final velocity ($v$).

$$P_{\text{final}} = (m_1 + m_2)v$$

By conservation of momentum, $P_{\text{initial}} = P_{\text{final}}$:

$$m_1 u_1 + m_2 u_2 = (m_1 + m_2)v$$ $$800 \text{ kg} \cdot \text{m/s} = (5 \text{ kg} + 25 \text{ kg})v$$ $$800 \text{ kg} \cdot \text{m/s} = (30 \text{ kg})v$$

Solve for $v$:

$$v = \frac{800 \text{ kg} \cdot \text{m/s}}{30 \text{ kg}}$$ $$v = \frac{80}{3} \text{ m/s} \approx 26.67 \text{ m/s}$$

$$KE_{\text{initial}} = \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2$$ $$KE_{\text{initial}} = \frac{1}{2}(5 \text{ kg})(10 \text{ m/s})^2 + \frac{1}{2}(25 \text{ kg})(30 \text{ m/s})^2$$ $$KE_{\text{initial}} = \frac{1}{2}(5 \text{ kg})(100 \text{ m}^2/\text{s}^2) + \frac{1}{2}(25 \text{ kg})(900 \text{ m}^2/\text{s}^2)$$ $$KE_{\text{initial}} = 250 \text{ J} + 11250 \text{ J}$$ $$KE_{\text{initial}} = 11500 \text{ J}$$

$$KE_{\text{final}} = \frac{1}{2}(m_1 + m_2)v^2$$ $$KE_{\text{final}} = \frac{1}{2}(30 \text{ kg})\left(\frac{80}{3} \text{ m/s}\right)^2$$ $$KE_{\text{final}} = \frac{1}{2}(30 \text{ kg})\left(\frac{6400}{9} \text{ m}^2/\text{s}^2\right)$$ $$KE_{\text{final}} = 15 \text{ kg} \times \frac{6400}{9} \text{ m}^2/\text{s}^2$$ $$KE_{\text{final}} = \frac{96000}{9} \text{ J}$$ $$KE_{\text{final}} = \frac{32000}{3} \text{ J} \approx 10666.67 \text{ J}$$

Loss in kinetic energy = $KE_{\text{initial}} - KE_{\text{final}}$

$$\text{Loss} = 11500 \text{ J} - \frac{32000}{3} \text{ J}$$ $$\text{Loss} = \frac{34500}{3} \text{ J} - \frac{32000}{3} \text{ J}$$ $$\text{Loss} = \frac{2500}{3} \text{ J}$$ $$\text{Loss} \approx 833.33 \text{ J}$$

The loss in kinetic energy is $\text{}\frac{2500}{3} \text{ J or } 833.33 \text{ J}*\text{}$