For shooting practice a person uses a pellet gun and an empty pop can. The pop can rests on a flat surface that has a coefficient of kinetic friction 0.500. The pellet has a mass of 0.140 kg and the pop can has a mass of 0.470 kg. After the pellet is fired and now inside the pop can they slide together for 6.80 m. What was the speed of the pellet?
total mass = .14 +.47 = .61 kg
weight = .61*9.81
friction force = .5 * .61 * 9.81
work done = force * distance = .5*.61*9.81*6.8
so
(1/2)(.61) v^2 = .5 * .61 * 9.81*6.8
where v is initial speed of can with pellet
v^2 = 9.81*6.8
solve for v
now conservation of momentum
u is pellet speed
.14 u = .61 v
To find the speed of the pellet, we can use the principle of conservation of momentum.
Step 1: Find the initial momentum of the pellet and the final momentum of the combined system.
The initial momentum of the pellet is given by:
Initial momentum = mass of the pellet * initial velocity of the pellet
Since the initial velocity is not given, we assume that the pellet starts from rest. Therefore, the initial momentum of the pellet is 0.
The final momentum of the combined system (pellet + pop can) can be calculated using the conservation of momentum:
Final momentum = (mass of the pellet + mass of the pop can) * final velocity of the combined system
Step 2: Find the final velocity of the combined system.
To find the final velocity, we can use the equation of motion:
Final velocity^2 = initial velocity^2 + 2 * acceleration * distance
Since the pellet and the pop can slide together, there is friction involved. The frictional force can be calculated using the equation:
Frictional force = coefficient of kinetic friction * normal force
The normal force acting on the pop can can be calculated using:
Normal force = mass of the pop can * acceleration due to gravity
Since the pop can is on a flat surface, the normal force is equal to the weight of the pop can:
Normal force = mass of the pop can * acceleration due to gravity
Therefore, the frictional force is:
Frictional force = coefficient of kinetic friction * mass of the pop can * acceleration due to gravity
The acceleration of the combined system can be calculated using Newton's second law of motion:
Frictional force = (mass of the pellet + mass of the pop can) * acceleration
Rearranging the equation, we get:
Acceleration = (coefficient of kinetic friction * mass of the pop can * acceleration due to gravity) / (mass of the pellet + mass of the pop can)
Now, substitute the value of acceleration into the equation for finding the final velocity.
Step 3: Calculate the final momentum and speed of the pellet.
The final momentum of the combined system is given by:
Final momentum = (mass of the pellet + mass of the pop can) * final velocity of the combined system
Since the pellet and the pop can are sliding together, their final velocities will be the same. Therefore, the final velocity of the pellet is the same as the final velocity of the combined system.
To find the speed of the pellet, we divide the final momentum of the pellet by its mass:
Speed of the pellet = Final momentum / mass of the pellet
Let's plug in the given values and calculate the final velocity and speed of the pellet:
To find the speed of the pellet, we can use the principle of conservation of momentum. The momentum of an object is defined as the product of its mass and velocity.
We are given the masses of the pellet and the pop can, as well as the distance they slide together. We can use this information to calculate the initial momentum of the pellet-can system before they slide.
Since the pop can is at rest initially, its momentum is zero. Thus, the initial momentum of the system is equal to the momentum of the pellet.
Let's assume the initial velocity of the pellet is u and the final velocity of the pellet-pop can system is v. According to the law of conservation of momentum, the momentum of the system before and after sliding must be the same.
So, we have:
Initial momentum = Final momentum
Mass of the pellet × Initial velocity of the pellet = (Mass of the pellet + Mass of the pop can) × Final velocity
Plugging in the given values:
0.140 kg × u = (0.140 kg + 0.470 kg) × v
Simplifying the equation:
0.140 kg × u = 0.610 kg × v
Now, we need to find the value of v, which is the final velocity of the pellet-can system.
To do this, we can use the concept of work-energy theorem. The work done on an object can be calculated by multiplying the force exerted on the object by the distance over which the force acts. In this case, the force is the force of friction.
The work done by the friction force can be calculated as follows:
Work done by friction = Force of friction × Distance
The force of friction can be calculated using:
Force of friction = Coefficient of kinetic friction × Normal force
The normal force is equal to the weight of the pellet-can system, which can be calculated as:
Normal force = Mass of the pellet + Mass of the pop can × gravitational acceleration
Now, let's calculate the normal force:
Normal force = (0.140 kg + 0.470 kg) × 9.8 m/s^2
Next, we calculate the force of friction:
Force of friction = 0.500 × Normal force
Finally, we can calculate the work done by the friction force:
Work done by friction = Force of friction × Distance
Since work done by friction is equal to the change in kinetic energy of the pellet-can system, we can say:
Work done by friction = Change in kinetic energy
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
The initial kinetic energy is given by:
Initial kinetic energy = (1/2) × Mass of the pellet × (initial velocity)^2
Since the pop can and the pellet start from rest, the initial kinetic energy is zero.
The final kinetic energy is given by:
Final kinetic energy = (1/2) × Total mass × (final velocity)^2
Total mass = Mass of the pellet + Mass of the pop can
Setting the work done by friction equal to the change in kinetic energy, we have:
Work done by friction = Final kinetic energy - Initial kinetic energy
Plugging in the given values:
Force of friction × Distance = (1/2) × (Mass of the pellet + Mass of the pop can) × (final velocity)^2
Simplifying the equation:
Force of friction × Distance = (1/2) × 0.610 kg × (final velocity)^2
Now that we have an equation relating the final velocity and the force of friction, we can substitute the known values in and solve for the final velocity.