Two smooth disks A and B have the initial velocities shown...

Two smooth disks $A$ and $B$ have the initial velocities shown just before they collide. If they have masses $m_{A}=4 \mathrm{kg}$ and $m_{B}=2 \mathrm{kg},$ determine their speeds just after impact. The coefficient of restitution is $e=0.8$.

Key Concepts

Conservation of Linear Momentum

This principle states that in a closed system with no external forces, the total momentum before a collision is equal to the total momentum after the collision. It is fundamental for solving collision problems by allowing the formulation of equations based on the mass and velocities of the objects involved, regardless of the nature of the collision.

Coefficient of Restitution

The coefficient of restitution is a measure of the elasticity of a collision between two bodies, defined as the ratio of the relative speed after collision to the relative speed before collision along the line of impact. It characterizes how much kinetic energy is conserved in the collision, with values ranging from 0 (perfectly inelastic collision) to 1 (perfectly elastic collision).

Transcript

00:02 In this problem, we have two smooth disks which are moving toward each other as shown.

00:07 They then collide and move apart.

00:10 We want to determine their speeds after impact.

00:13 We choose our y and x axes as follows.

00:17 So the y -axis is along the line of impact.

00:22 Now, we can write our velocities, our known velocities as follows.

00:30 So v -a -y -1, where subscript 1 denotes before the impact.

00:36 Is equal to 15 times 3 over 5.

00:43 And so along the y -axis, this is 9 meters per second in that direction.

00:55 Now along the x -axis, v -a -x -1 is equal to 15 multiplied by 4 over 5, and this is 12 meters per second to the northwest.

01:16 So we have our x and y components of the initial velocity of a.

01:21 For b, we do the same.

01:24 So vb x1 before the collision is eight meters per second purely along the x -axis.

01:39 And that's not the x component, that is the y component.

01:49 The x component vb x1 is equal to 0.

01:55 So we see b moves purely along the y -axis.

01:59 So the x component of its velocity initially is zero.

02:04 Now that we have the velocities, let's apply the coefficient of restitution.

02:15 So we know the coefficient of restitution, taking the right direction along our y -axis to be positive.

02:23 The coefficient of restitution e is equal to vb y2 minus vayy 2 over vayy 1 minus vb y1 y1 and we know this coefficient of restitution as it is given the problem to be 0 .8 so this is equal to vb y2 after the collision minus v a y2 these are both unknown over minus 9 minus 8.

03:17 So we get an equation here...

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