Two football players are practicing their tackles. Fred (mass 60 kg) runs to the right at 5.4 m/s. Barney (115 kg) runs towards him straight-on and tackles him. They fall to the ground and slide 47 cm to the left. If the coefficient of kinetic friction between the ground and their uniforms is 0.11, what was Barney’s speed as he ran toward Fred
I hardly call this an elastic collision. With that said, assuming it was elastic, and all of the final KE was dissipated by friction force on the ground...
Initial KE=final work
1/2 *60*g*5.4^2 +1/2*115g*V^2=(115+60)g*.11*.47
divide the g's out, solve for V
To find Barney's speed as he ran toward Fred, we can make use of the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of a system remains constant before and after a collision, assuming no external forces act on the system.
We know the mass and initial velocity of Fred, and the final displacement of both players. We can set up the equation as follows:
Initial momentum of Fred = Final momentum of Fred + Final momentum of Barney
The initial momentum of Fred is given by:
Initial momentum of Fred = mass of Fred * velocity of Fred
The final momentum of Fred is given by:
Final momentum of Fred = mass of Fred * final velocity of Fred
The final momentum of Barney is given by:
Final momentum of Barney = mass of Barney * final velocity of Barney
Since they slide to the left after the collision, we can consider the final velocity of Fred and Barney as negative quantities.
Putting it all together and rearranging the equation, we have:
mass of Fred * velocity of Fred = mass of Fred * final velocity of Fred + mass of Barney * final velocity of Barney
Substituting the given values:
60 kg * 5.4 m/s = 60 kg * (final velocity of Fred) + 115 kg * (final velocity of Barney)
Since they slide 47 cm to the left, we convert it to meters by dividing by 100:
47 cm = 0.47 m (final displacement)
Now we can use the equation of motion that relates final displacement, initial velocity, final velocity, and acceleration:
(final velocity)^2 = (initial velocity)^2 + 2 * acceleration * final displacement
When sliding, the only force acting on Fred and Barney is friction, so we can find the acceleration using the equation:
frictional force = coefficient of kinetic friction * normal force
frictional force = mass * acceleration
Since the normal force is equal to the weight (mass * gravity) when they are on a horizontal surface, we have:
frictional force = coefficient of kinetic friction * mass * gravity
mass * acceleration = coefficient of kinetic friction * mass * gravity
Canceling out the mass on both sides, we get:
acceleration = coefficient of kinetic friction * gravity
Substituting the given values:
acceleration = 0.11 * 9.8 m/s^2
Now that we have the acceleration, we can solve for final velocity using the equation of motion:
(final velocity)^2 = (initial velocity)^2 + 2 * acceleration * final displacement
(final velocity of Fred)^2 = (5.4 m/s)^2 + 2 * (0.11 * 9.8 m/s^2) * 0.47 m
Solving for (final velocity of Fred)^2:
(final velocity of Fred)^2 = 29.16 m^2/s^2 + 2.14 m^2/s^2
(final velocity of Fred) ≈ √31.30 m/s ≈ 5.59 m/s (rounding to two decimal places)
Now we can substitute this value back into the conservation of momentum equation to solve for the final velocity of Barney:
60 kg * 5.4 m/s = 60 kg * 5.59 m/s + 115 kg * (final velocity of Barney)
Simplifying the equation:
324 kg m/s = 335.40 kg m/s + 115 kg * (final velocity of Barney)
Rearranging the equation to solve for (final velocity of Barney):
115 kg * (final velocity of Barney) = 324 kg m/s - 335.40 kg m/s
(final velocity of Barney) = (324 kg m/s - 335.40 kg m/s) / 115 kg
Simplifying the equation to find the final velocity of Barney:
(final velocity of Barney) ≈ -0.10 m/s
Therefore, Barney's speed as he ran toward Fred was approximately 0.10 m/s in the opposite direction.