A sphere is rolling towards a stationary inclined wedge (inclination 37 degrees) with the speed of 10m/s. The mass of sphere and wedge are equal to 5kg. assuming no friction and e=0.2. Find the final velocities of the objects.

视频信息

答案文本 复制

视频字幕 复制

We have a collision problem involving a sphere and an inclined wedge. A sphere with mass 5 kilograms is rolling toward a stationary wedge, also with mass 5 kilograms, at an initial velocity of 10 meters per second. The wedge has an inclination angle of 37 degrees. The coefficient of restitution is 0.2, and we assume frictionless surfaces. We need to find the final velocities of both objects after the collision. To solve this collision problem, we need to apply three fundamental conservation principles. First, we have conservation of momentum in the horizontal direction, which states that the total horizontal momentum before collision equals the total horizontal momentum after collision. Second, we have conservation of momentum in the vertical direction with the same principle. Third, we use the coefficient of restitution, which is 0.2 in this case, to relate the relative velocities before and after collision along the normal direction. These three equations will allow us to solve for the three unknown final velocities of both objects. Now we need to analyze the velocity components during the collision. Initially, the sphere has a horizontal velocity of 10 meters per second and zero vertical velocity, while the wedge is stationary. To properly analyze the collision, we must decompose the velocities into components normal and tangential to the inclined surface. The normal direction is perpendicular to the inclined surface, while the tangential direction is parallel to it. Using the 37-degree angle, we can calculate that the sphere's initial normal component is 10 times sine of 37 degrees, which equals 6 meters per second, and the tangential component is 10 times cosine of 37 degrees, which equals 8 meters per second. Now we solve the system of equations step by step. First, we apply horizontal momentum conservation: the initial momentum of 50 kilogram meters per second equals the sum of final horizontal momenta, giving us 10 equals v1fx plus v2fx. Second, vertical momentum conservation gives us zero equals v1fy plus v2fy, since there was no initial vertical momentum. Third, the coefficient of restitution equation becomes 1.2 equals v2n minus v1n, where the normal components must be converted to x-y coordinates using the 37-degree angle. After solving this system of three equations with three unknowns, we get the final velocities: sphere horizontal 2.56 meters per second, sphere vertical negative 1.92 meters per second, wedge horizontal 7.44 meters per second, and wedge vertical 1.92 meters per second. Here are the final results of our collision analysis. After the collision, the sphere has a final velocity of 3.20 meters per second, moving rightward at 2.56 meters per second and downward at 1.92 meters per second. The wedge moves with a final velocity of 7.69 meters per second, rightward at 7.44 meters per second and upward at 1.92 meters per second. We can verify our solution by checking momentum conservation: the initial momentum of 50 kilogram meters per second equals the final momentum. The kinetic energy decreases from 250 joules to 173 joules, with 77 joules lost due to the inelastic nature of the collision, as indicated by the coefficient of restitution being less than one. This energy loss is expected in real collisions and confirms our solution is physically reasonable.