A dog of mass 8kg chases a bicycle rider at a speed of 20ms' . The mass of the rider and the bicycle is 48kg and in moving ata speed of 5m-1. if the dog rams and stricks into the spokes of the bicycle, find their common velocity after, and the type of the collision.
We can solve this problem using conservation of momentum:
Initial momentum = Final momentum
The initial momentum is the sum of the momentum of the dog and the momentum of the rider and bicycle:
Initial momentum = (8 kg)(20 m/s) + (48 kg)(5 m/s)
Initial momentum = 160 kg m/s + 240 kg m/s
Initial momentum = 400 kg m/s
After the collision, the dog and the rider and bicycle will move together at a common velocity. We can call this final velocity v.
The final momentum is the momentum of the combined system:
Final momentum = (8 kg + 48 kg)(v)
Final momentum = 56 kg(v)
Equating the initial and final momenta:
Initial momentum = Final momentum
400 kg m/s = 56 kg(v)
v = 7.14 m/s
Therefore, the common velocity after the collision is 7.14 m/s.
To determine the type of collision, we need to look at whether kinetic energy is conserved in the collision. If kinetic energy is conserved, the collision is elastic. If kinetic energy is not conserved, the collision is inelastic.
To find out if kinetic energy is conserved, we can calculate the initial kinetic energy and the final kinetic energy:
Initial kinetic energy = (1/2)(8 kg)(20 m/s)^2 + (1/2)(48 kg)(5 m/s)^2
Initial kinetic energy = 1600 J + 600 J
Initial kinetic energy = 2200 J
Final kinetic energy = (1/2)(56 kg)(7.14 m/s)^2
Final kinetic energy = 1429 J
Since the initial kinetic energy is greater than the final kinetic energy, we know that kinetic energy is not conserved in the collision. Therefore, the collision is inelastic.
To solve this problem, we can use the principle of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision.
First, let's calculate the initial momentum of the dog and the bicycle rider:
Momentum of the dog (Pdog) = mass of the dog (mdog) × velocity of the dog (vdog)
Pdog = 8 kg × 20 m/s
Pdog = 160 kg·m/s
Momentum of the bicycle rider (Prider+bicycle) = mass of the rider+bicycle (mrider+bicycle) × velocity of the rider+bicycle (vrider+bicycle)
Prider+bicycle = (48 kg + 8 kg) × 5 m/s
Prider+bicycle = 56 kg × 5 m/s
Prider+bicycle = 280 kg·m/s
The total initial momentum (Pinitial) is the sum of the dog's momentum and the bicycle rider's momentum:
Pinitial = Pdog + Prider+bicycle
Pinitial = 160 kg·m/s + 280 kg·m/s
Pinitial = 440 kg·m/s
Now, let's consider the collision between the dog and the bicycle. Since the dog rams and strikes into the spokes of the bicycle, we can assume this is an inelastic collision. In an inelastic collision, the two objects stick together after colliding.
Let the common velocity of the dog and the bicycle after the collision be vf.
According to the principle of conservation of momentum:
Pinitial = Pfinal
440 kg·m/s = (mass of dog + mass of rider+bicycle) × vf
Substituting the masses:
440 kg·m/s = (8 kg + 48 kg) × vf
440 kg·m/s = 56 kg × vf
Dividing both sides of the equation by 56 kg:
7.85 m/s = vf
Therefore, the common velocity of the dog and the bicycle after the collision is approximately 7.85 m/s.
Since the two objects stick together after colliding, the type of collision is an inelastic collision.