A firecracker travels straight up at 75.0 m/s and explodes into two pieces. The smaller piece carries one third of the firecracker’s mass and shoots off horizontally at 48.0 m/s west. Determine the speed and direction of the other piece directly after the explosion.

momentum is conserved , vertically and horizontally

the larger piece carries all of the vertical momentum , and horizontal momentum equal and opposite to the smaller piece

find the vertical and horizontal momentum vectors and add them

we will be happy to critique your work.

To determine the speed and direction of the other piece of the firecracker directly after the explosion, we can use the law of conservation of momentum. According to this law, the total momentum before the explosion should be equal to the total momentum after the explosion.

Let's assume the mass of the firecracker is "m" and the mass of the smaller piece is "m/3". The velocity of the firecracker before the explosion is 75.0 m/s straight up.

The total initial momentum before the explosion is given by:

Initial momentum = Mass of firecracker * Velocity of firecracker

P initial = m * 75.0 m/s

Since the smaller piece carries one third of the firecracker's mass, its mass is m/3. It shoots off horizontally at 48.0 m/s west, which means its velocity has both magnitude and direction.

The initial momentum of the smaller piece is given by:

Initial momentum of smaller piece = Mass of smaller piece * Velocity of smaller piece

P initial(smaller piece) = (m/3) * (-48.0 m/s) (Negative sign indicates westward direction)

According to the law of conservation of momentum, the total initial momentum before the explosion is equal to the total momentum after the explosion.

P initial = P initial(smaller piece) + P initial (larger piece)

m * 75.0 m/s = (m/3) * (-48.0 m/s) + P initial (larger piece)

Now, you have an equation with one variable, "m", which represents the mass of the firecracker. We can solve this equation to find the value of "m".

m * 75.0 m/s = (m/3) * (-48.0 m/s) + P initial (larger piece)

Simplifying the equation:

75.0 m = -16.0 m + P initial (larger piece)

P initial (larger piece) = 75.0 m + 16.0 m

P initial (larger piece) = 91.0 m

Therefore, the initial momentum of the larger piece directly after the explosion is 91.0 m.

The speed of an object is given by the magnitude of its velocity. Since the smaller piece's velocity is given as 48.0 m/s, its speed is 48.0 m/s.

Now, to find the speed and direction of the other piece directly after the explosion, we need to find its velocity.

The final momentum of the smaller piece is given by:

Final momentum = Mass of smaller piece * Velocity of smaller piece

P final(smaller piece) = (m/3) * (-48.0 m/s)

According to the law of conservation of momentum, the total final momentum after the explosion is equal to the total initial momentum before the explosion.

P initial = P final(smaller piece) + P final(larger piece)

91.0 m = (m/3) * (-48.0 m/s) + P final(larger piece)

Now, you have an equation with one variable, "m", which represents the mass of the firecracker. We can solve this equation to find the value of "m".

91.0 m = (m/3) * (-48.0 m/s) + P final(larger piece)

Simplifying the equation:

48.0 m = P final(larger piece)

Therefore, the momentum of the larger piece directly after the explosion is 48.0 m.

Since momentum is the product of mass and velocity, we can write:

P = m * v

48.0 m = m * v

Simplifying the equation:

v = 48.0 m/s

So, the speed of the other piece directly after the explosion is 48.0 m/s. Since we know that this piece moves in the opposite direction of the smaller piece, its direction can be determined as east.

Therefore, the speed and direction of the other piece directly after the explosion are 48.0 m/s east.