Rockfalls can cause major damage to roads and...

Rockfalls can cause major damage to roads and infrastructure. To design mitigation bridges and barriers, engineers use the coefficient of restitution to model the behavior of the rocks. Rock $A$ falls a distance of $20 \mathrm{~m}$ before striking an incline with a slope of $a=40^{\circ} .$ Knowing that the coefficient of restitution between rock $A$ and the incline is 0.2 , determine the velocity of the rock after the impact.

Key Concepts

Gravitational Potential Energy Conversion

This concept refers to the transformation of potential energy into kinetic energy as an object falls under gravity. In free fall, the potential energy lost due to a change in height is converted into kinetic energy, which determines the speed of the falling object just before impact.

Coefficient of Restitution

The coefficient of restitution is a measure of the elasticity of a collision between two surfaces. It is defined as the ratio of the relative speed after collision to the relative speed before collision along the normal (perpendicular) to the surfaces at the point of impact. This value indicates how much kinetic energy is conserved in the collision, with values closer to 1 representing nearly elastic collisions and values closer to 0 representing highly inelastic collisions.

Vector Decomposition in Collisions

When an object impacts an inclined surface, its velocity can be resolved into two components: one perpendicular (normal) and one parallel (tangential) to the surface. The coefficient of restitution primarily affects the normal component of the velocity, reducing it based on the elasticity of the collision, while the tangential component often remains unchanged if friction is neglected. This decomposition is essential for accurately modeling the post-impact velocity direction and magnitude.

Transcript

00:02 So firstly we'll draw a diagram of our falling rock.

00:07 So the rock falls from a distance y above its point of impact.

00:13 It falls down vertically with initial velocity vb.

00:18 The angle between the slope and the datum is alpha.

00:26 The rock then rebounds and has final velocity v prime.

00:32 The v prime, the final velocity can be broken up into its n and t components.

00:39 So the n component along the n axis is vb prime, n and along the t axis is vb prime t.

00:48 We are given that y is 20 meters.

00:51 The coefficient of restitution at the point of impact is 0 .2 and alpha is equal to 40 degrees.

00:58 So we've also labeled the angle beta to be the angle between the vector vb and its t component.

01:10 So the first thing we will do is apply the work energy theorem.

01:14 So the work energy theorem will help us determine the velocity of the rock on impact.

01:22 So work energy tells us that the initial kinetic energy plus the initial gravitational potential energy plus the initial elastic potential energy when the rocket is at its highest point, plus the external work done in going from the first to the second state, or in this case the third state, after the impact, must equal to the final kinetic energy when the rock has rebounded, plus its gravitational potential energy, plus its elastic potential energy.

02:03 So initially the rock has gravitational potential energy and we assume it's moving from rest as it falls off from a higher position.

02:13 There's no elastic potential energy and there's no friction or any other non -conservative external forces acting on the system.

02:23 Finally, the rock just before impact has only kinetic energy.

02:27 So basically all this potential gravitational potential energy is converted to kinetic energy when the rock reaches.

02:34 The datum so vg y so vg 1 is actually m g times the height above the point of impact y and this is equal to t2 and t2 is equal to half times the mass of the rock times its speed vb squared so if you rearrange this equation and we substitute our values in we get that the speed of b at the point impact is the square root of 40 times acceleration into gravity g and that's meters per second.

03:19 So now we have the linear speed that the rock is moving with.

03:26 So now we take n and t components and the n component of the velocity vb is equal to minus vb cost alpha the t component of vb, vbt is equal to vb sine alpha.

04:06 So vb is the velocity here of the rock when it reaches the point of impact...

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