Two ice skaters, Daniel (mass 65.0 kg ) and Rebeca (mass 45.0 kg , are practicing. Daniel stops

Two ice skaters, Daniel (mass 65.0 kg ) and Rebeca (mass...

Two ice skaters, Daniel (mass 65.0 $\mathrm{kg}$ ) and Rebeca (mass 45.0 $\mathrm{kg}$ , are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 $\mathrm{m} / \mathrm{s}$ before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 $\mathrm{m} / \mathrm{s}$ at an angle of $53.1^{\circ} \mathrm{from}$ her initial direction. Both skaters move on the frictionless, horizontal surface of the rink. (a) What are the magnitude and direction of Daniel's velocity after the collision? (b) What is the change in total kinetic energy of the two skaters as a result of the collision?

Two ice skaters, Daniel (mass 65.0 $\mathrm{kg}$ ) and Rebeca (mass 45.0 $\mathrm{kg}$ , are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 $\mathrm{m} / \mathrm{s}$ before she collides with him. After the collision, Rebecca has a velocity of
magnitude 8.00 $\mathrm{m} / \mathrm{s}$ at an angle of $53.1^{\circ} \mathrm{from}$ her initial direction. Both skaters move on the frictionless, horizontal surface of the rink. (a) What are the magnitude and direction of Daniel's velocity after the collision? (b) What is the change in total kinetic energy of the two skaters as a result of the collision?

Key Concepts

Kinetic Energy Analysis

While momentum is always conserved in collisions, kinetic energy is conserved only in elastic collisions and not in inelastic collisions. Analyzing the change in kinetic energy before and after a collision reveals the energy lost or transformed during the impact, which is crucial for understanding the collision dynamics.

Conservation of Momentum

This principle states that in the absence of external forces, the total momentum of a system remains constant before and after a collision. In problems involving collisions, momentum must be conserved in each direction separately, which often requires setting up equations for both the horizontal and vertical components.

Vector Decomposition

In two-dimensional collisions, velocities are often given at angles, necessitating the use of trigonometry to decompose these vectors into perpendicular components (typically horizontal and vertical). This enables the independent application of conservation laws along each axis.

Transcript

00:01 Here in this question we have given mass of person 1 that is m1 equals to 65 kg mass of person 2 that is m2 equal to 45 kg also speed before collision that is u1 equal to 0 and u2 equal to 13 meter per second and speed after collision.

01:06 Here we have the speed of second person that is v2 equal to 8 meter per second and also we have the value of of theta equal to 53 .1 degree.

01:26 Now we have to find out the magnitude of velocity and direction of daniel after collision and then we have to calculate the change in total kinetic energy.

01:45 So here we know that according to low of conservation of conservation of momentum if x and y component of total momentum so it gives m1 u1 x plus m2 u2 x equal to m1 v1 x plus m2 v2 x and similarly, m1u1y plus m2u2y equals to m1v1y plus m1v2y plus m2v2y.

02:51 Now we consider this as equation.

02:58 Now here we assume that the speed and direction of the person 1 after collision b, v1, and theta respectively so we have u1 x equal to u1y equal to 0 u2 x equal to 13 meter per second u2y equal to 0 v1 x equal to v1 cos theta and v1 1y equal to v1 sine theta and v2 x equal to 8 course 53 .1 degree also v2y equal to 8 sine 53 .1 degree which is equal to 6 .4 meter per second and v2x is actually equal to 4 .8 meter per second.

04:35 So now when we putting these in the equation 1 and equation 2, we have v1 x equal to m2 multiply by u2x minus v2x divided by m1 now we substitute the value so we get v1 x equal to 45 multiply by 13 minus 4 .8 divided by 65 so we get v1 x equal to 5 .7 meter per second.

05:32 Now similarly v1 y from equation 2 can be written as minus m2 divided by m1 multiply by v2 y now here on putting the values we get v1 y equals to minus 45 divided by 65 multiply by 6 .4 so we get v1y equals to minus 4 .43 meter per second now we are going to find the magnitude of velocity for daniel and then find the direction also so so here in part a, magnitude of velocity is given by v1 equal to square root of v1x square plus v1y square.

07:00 Now on putting the values we get v1 equal to square root of 5 .7 whole square plus minus 4 .43 whole square.

07:17 So we get v1 equals to 7 .2 meter per second.

07:28 So this is the magnitude of velocity for daniel after the collision.

07:34 Now we find the direction...

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