91Ó°ÊÓ

To solve this, we will use two principles: Firstly, the principle of energy conservation, which states that total mechanical energy in a system is conserved if the only forces doing work are conservative forces. In this problem the crate is affected by nonconservative forces due to friction. Because of that we need to use a modified form of the principle of energy conservation that includes work done by nonconservative forces such as friction. Secondly we will use the friction force equation to calculate work done by friction.So, let's write the conservation of energy principle which accounts for non-conservative work: \[KE_{A} + PE_{A} + W_{nc} = KE_{B} + PE_{B}\]Where:- \( KE \) is the kinetic energy given by \( 1/2 mv^2 \),- \( PE \) is the potential energy given by \( mgh \), where \( h \) is the height,- \( m \) is the mass of the object, \( v \) is the velocity of the object, - \( g \) is the acceleration due to gravity, and - \( W_{nc} \) is the work done by non-conservative forces. We set point \( A \) just before it starts moving (higher potential energy, initial kinetic energy) and point \( B \) just before it leaves the ramp (lower potential energy, final kinetic energy). All variables related to points \( A \) and \( B \) will be subscripted by \( A \) and \( B \). Energy conservation then gives us:\[\frac{1}{2} m {v_A}^2 + mgh_A - F_{friction} * d = \frac{1}{2} m {v_B}^2 + mgh_B\]From the given problem, the velocity at \( A \) is \(2.5 \) m/s and at \( B \) it is what we are trying to find. The height at \( A \) can be derived by simple trigonometry to be \( d(sin \theta)\), as \( d \) is the length of the ramp and \( \theta \) is the angle of inclination, while at \( B \) it will be \( 0 \) (as \( B \) is the bottom). \( F_{friction} \) can be calculated as \( \mu_k * F_{normal} \), where \( F_{normal} \) is the normal force which in this case would be \( mg cos \theta \). \( d \) is the distance traveled.

First we determine the friction force. For this, the friction equation is used:\[F_{friction} = \mu_k * F_{normal}\]By substituting the given values into this equation we get:\[F_{friction} = 0.3 * (12 * 9.81 * cos(30))\]Then we can substitute all known values into the conservation of energy equation. Because the initial velocity and the final height we seek are unknown, the equation becomes:\[\frac{1}{2} * 12 * (2.5)^2 + 12 * 9.81 * d * sin(30) - F_{friction} * d = \frac{1}{2} * 12 * {v_B}^2 + 0\]It is assumed that d cancels on both sides. Now, almost all values are known, and the equation can be solved for \( v_{B}\).After rearranging and solving we get the velocity at \( B \), \( v_{B} \), in m/s.