You shoot your enemy's reflection in the mirror while playing paintball. The y axis is perpendicular to the mirror. The mirror has a mass of 1kg and the paintball has a mass of 0.05 kg. The paintball hits the mirror at a speed of 8 m/s at an angle of 25 degrees from the normal. The mirror is pushed backward upon the impact remaining facing the same direction it was before. Assuming that the collision was perfectly elastic, find the final velocity of the paintball. Does it hit your enemy?
v=6.78
To find the final velocity of the paintball, we can use the principle of conservation of momentum. The total momentum before the collision is equal to the total momentum after the collision.
Initially, the mirror is at rest, so its momentum is zero. The paintball's momentum before the collision can be calculated using its mass and velocity. Since the angle of 25 degrees is given with respect to the normal, we will first find the component of velocity perpendicular to the mirror and the component parallel to the mirror.
Perpendicular component of paintball velocity:
v_perpendicular = 8 m/s * sin(25 degrees)
Parallel component of paintball velocity:
v_parallel = 8 m/s * cos(25 degrees)
Now, let's calculate the initial momentum of the paintball:
initial momentum of paintball = mass of paintball * v_parallel
After the collision, the mirror moves backward, acquiring a velocity. Let's denote this final velocity of the mirror as v_mirror. The paintball moves forward with a final velocity, which we need to find.
Using the law of conservation of momentum:
initial momentum of paintball + initial momentum of mirror = final momentum of paintball + final momentum of mirror
Since the mirror is initially at rest, the initial momentum of mirror is zero:
initial momentum of paintball = final momentum of paintball + final momentum of mirror
Substituting the values:
mass of paintball * v_parallel = mass of paintball * final velocity of paintball + mass of mirror * final velocity of mirror
We are given the mass of the paintball and mirror. The only unknowns are the final velocity of the paintball and mirror. However, we also know something about the final velocity of the mirror:
"The mirror is pushed backward upon the impact, remaining facing the same direction it was before."
This means that the final velocity of the mirror will be in the negative direction, opposite to the positive direction of the paintball. Let's denote the final velocity of the paintball as v_pb and the final velocity of the mirror as -v_m, where v_m is the magnitude of final mirror velocity.
mass of paintball * v_parallel = mass of paintball * v_pb + mass of mirror * (-v_m)
Now, we can solve for v_pb:
v_pb = (mass of paintball * v_parallel - mass of mirror * (-v_m)) / mass of paintball
Knowing the values of v_parallel, v_perpendicular, mass of paintball, mass of mirror, and v_m, you can substitute them into the equation to find v_pb.
Regarding whether the paintball hits your enemy, we would need information about the distance between the mirror and your enemy and the time it takes for the paintball to travel. That information is not provided in the question, so we cannot determine if the paintball hits your enemy.