A 0.020-kg bullet is shot horizontally and collides with a \(2.00-\mathrm{kg}\) block of wood. The bullet embeds in the block and the block slides along a horizontal surface for \(1.50 \mathrm{m} .\) If the coefficient of kinetic friction between the block and surface is \(0.400,\) what was the original speed of the bullet?

Before the collision, we have the bullet moving with an initial velocity (v_bullet_initial) and the wooden block at rest. After the collision, both bullet and block move together with a combined velocity (v_combined). We can use the conservation of momentum to find the combined velocity. Momentum_before = Momentum_after (m_bullet * v_bullet_initial) + (m_block * 0) = (m_bullet + m_block) * v_combined Now we can solve the equation for v_combined: v_combined = (m_bullet * v_bullet_initial) / (m_bullet + m_block)

Now that we have v_combined, we can use the work-energy principle to relate the work done by friction to the change in kinetic energy: Work_done = Change_in_Kinetic_Energy (Force_friction * distance) = (1/2 * (m_bullet + m_block) * v_combined^2) - (1/2 * (m_bullet + m_block) * 0^2) The force of friction can be found using the equation: Force_friction = coefficient_of_friction * Normal_Force Since the block is on a horizontal surface, the normal force is equal to the gravitational force acting on the block: Normal_Force = (m_bullet + m_block) * g Substituting Normal_Force in the Work_done equation, we get: Work_done = -coefficient_of_friction * (m_bullet + m_block) * g * distance We used a negative sign before the coefficient_of_friction because the friction force opposes the motion of the block. Now, plug in the given values and solve for v_bullet_initial: -0.400 * (0.020 + 2.00) * 9.81 * 1.50 = (1/2 * (0.020 + 2.00) * v_combined^2)

Now we have two equations: v_combined = (0.020 * v_bullet_initial) / (0.020 + 2.00) and -0.400 * (0.020 + 2.00) * 9.81 * 1.50 = (1/2 * (0.020 + 2.00) * v_combined^2) First, solve the second equation for v_combined: v_combined^2 = (-0.400 * (0.020 + 2.00) * 9.81 * 1.50) / (0.5 * (0.020 + 2.00)) Now substitute the first equation into the second equation to find the initial speed of the bullet: v_bullet_initial = ((0.020+2.00) * (-0.400 * (0.020 + 2.00) * 9.81 * 1.50) / (0.5 * (0.020 + 2.00))) / 0.020 Performing the calculation, we obtain: v_bullet_initial ≈ 167.76 m/s So, the initial speed of the bullet was approximately 167.76 m/s.