A ball of mass 2m is traveling EAST with a velocity of 2v. A ball of m is traveling NORTH with a velocity of 3v. The balls stick together (they are both made out of putty). How fast are they going immediately after they stick together? In what direction are they going immediately after they stick together?
momentum east = 4 v m
momentum north = 3 v m
final mass = 7 m
final v east = 4 v m/7m = (4/7) v
final v north = 3 v m/7m = (3/7) v
speed magnitude = (v/7)sqrt (16+9)
T = angle above x (east) axis
tan T = 3/4
T = 36.9 deg and thus angle clockwise from north = 53.1 on gyrocompass
Given:
M1 = 2m, V 1 = 2v.
M2 = m, V2 = 3v.[90o].
V3 = Velocity of M1 and M2 after collision.
Momentum before = Momentum after:
M1*V1 + M2*V2 = M1*V3 + M2*V3.
2m*2v + m*3vi = 2m*V3 + m*V3,
Divide both sides by m:
4v + 3vi = 3V3,
3V3 = 5v[36.9o],
V3 = 1.67v[36.9o].
Tan A = 3v/4v = 0.75, A = 36.9o.
To solve this problem, we can use the principles of conservation of momentum. According to the law of conservation of momentum, the total momentum of an isolated system before an event is equal to the total momentum after the event.
Step 1: Determine the initial momentum of each ball:
The momentum of an object is given by the equation p = mv, where p is the momentum, m is the mass, and v is the velocity.
For the first ball with a mass of 2m and traveling east with a velocity of 2v, the momentum is given by:
P1 = (2m)(2v) = 4mv (momentum of the first ball)
For the second ball with a mass of m and traveling north with a velocity of 3v, the momentum is given by:
P2 = (m)(3v) = 3mv (momentum of the second ball)
Since the balls stick together, their total momentum after sticking together will be equal to the sum of their individual momenta before sticking together.
Step 2: Determine the total momentum after the balls stick together:
P_total = P1 + P2 = 4mv + 3mv = 7mv
The total momentum after the balls stick together is 7mv.
Step 3: Determine the final velocity of the combined system:
The final velocity can be calculated by dividing the total momentum by the combined mass of the two balls.
The combined mass is the sum of the individual masses:
m_total = 2m + m = 3m
The final velocity (V_final) is given by:
V_final = P_total / m_total = (7mv) / (3m) = 7v/3
So, the combined balls will be traveling at a speed of 7v/3 after sticking together.
Step 4: Determine the direction:
To find the direction, we need to consider the various velocities and their directions. The first ball is traveling east, while the second ball is traveling north.
Since the final velocity (V_final) is calculated as a scalar value (magnitude only), we can consider the angle between the initial velocity vectors of the two balls. In this case, the angle is 90 degrees because one ball is traveling east, and the other is traveling north.
Thus, the combined balls will be moving at a speed of 7v/3 and in the direction of the resultant vector, which is the diagonal formed by the east and north directions.
To specify the direction, we can describe it as 45 degrees northeast (relative to the original reference frame).
Therefore, immediately after sticking together, the balls will be traveling at a speed of 7v/3 and in the direction of 45 degrees northeast.