A basketball explodes into three separate pieces. Two pieces of equal mass shoot off at 20.3 m/s, one travelling to the south and one to the west, perpendicular to each other. The third piece has a mass 4 times the size of one of the other pieces. What is the speed of the third piece?
Could you actually show me how to do it?
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the explosion should be equal to the total momentum after the explosion.
Let's assume the mass of each of the smaller pieces before the explosion is m kg. Thus, the mass of the third piece is 4m kg.
Before the explosion, the basketball is at rest, so its momentum is zero.
After the explosion, the two smaller pieces shoot off perpendicular to each other. Let's denote the velocity of the piece traveling south as v_s and the velocity of the piece traveling west as v_w.
Since the two smaller pieces have equal mass and velocity, their momenta are the same. Therefore, we have:
momentum of the piece traveling south (p_s) = mass of the piece * velocity of the piece
p_s = m * v_s
momentum of the piece traveling west (p_w) = mass of the piece * velocity of the piece
p_w = m * v_w
The total momentum after the explosion is the vector sum of the individual momenta:
total momentum = p_s + p_w
Using the Pythagorean theorem, we can calculate the magnitude of the total momentum. The total momentum must be zero because the basketball was originally at rest:
0 = √((m * v_s)^2 + (m * v_w)^2)
Simplifying the equation:
0 = m^2 * (v_s^2 + v_w^2)
So, v_s^2 + v_w^2 = 0
We know that v_s = 20.3 m/s and v_w = 20.3 m/s because the two smaller pieces have equal velocity.
Substituting these values into the equation:
(20.3)^2 + (20.3)^2 = 0
412.09 + 412.09 = 0
824.18 = 0
The equation is not satisfied, meaning there must be an error in the problem statement. It's not possible for the third piece to have a mass 4 times the size of the other pieces and for it to have a speed.
To find the speed of the third piece, we first need to determine the initial momentum of the basketball before it exploded. Momentum is defined as:
Momentum = Mass × Velocity
Since there are three pieces after the explosion, we can use the principle of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event.
Before the explosion, the basketball was initially at rest, so its momentum was zero. After the explosion, the two smaller pieces shoot off at 20.3 m/s each, in perpendicular directions (south and west).
Let's denote the mass of each smaller piece as m, and the mass of the larger piece as 4m (4 times the size of one of the other pieces).
To find the total momentum after the explosion, we calculate the momentum of each piece and add them up.
Momentum of the first small piece (going south):
Momentum1 = Mass × Velocity = m × 20.3 m/s (south)
Momentum of the second small piece (going west):
Momentum2 = Mass × Velocity = m × 20.3 m/s (west)
Momentum of the larger piece:
Momentum3 = Mass × Velocity
Since the mass is 4 times larger than each of the smaller pieces, the velocity of the larger piece will be denoted as v.
Momentum3 = (4m) × v
The total momentum after the explosion is the sum of the individual momenta:
Total Momentum = Momentum1 + Momentum2 + Momentum3
Since momentum is a vector quantity, we need to use vector addition to combine the individual momenta. Since the two smaller pieces are at right angles to each other, we can use the Pythagorean theorem to calculate their combined magnitude:
Magnitude of the total momentum = √(Momentum1^2 + Momentum2^2)
Now we can set up the equation and solve for the total momentum:
√(Momentum1^2 + Momentum2^2) + Momentum3 = 0
Substituting in the given values:
√((m × 20.3 m/s)^2 + (m × 20.3 m/s)^2) + (4m) × v = 0
Simplifying:
√(2 × m^2 × (20.3 m/s)^2) + 4m × v = 0
2 × m × 20.3 m/s + 4m × v = 0
40.6 m × m/s + 4m × v = 0
Now we need to solve for the velocity of the third piece (v):
4m × v = -40.6 m × m/s
v = (-40.6 m × m/s) / (4m)
Simplifying:
v = -10.15 m/s
Therefore, the speed of the third piece is 10.15 m/s in the opposite direction of momentum2 (west in this case).