An m = 6.80-kg clay ball is thrown directly against a perpendicular brick wall at a velocity of 22.0 m/s and shatters into three pieces, which all fly backward, as shown in the figure. The wall exerts a force on the ball of 2670 N for 0.110 s. One piece of mass m1 = 2.80 kg travels backward at a velocity of 10.1 m/s and an angle of = 32.0° above the horizontal. A second piece of mass m2 = 1.80 kg travels at a velocity of 8.30 m/s and an angle of 28.0° below the horizontal.

I found the velocity of the third piece to be 48.738 m/s

but i need help finding:
What is the direction of the third piece?

good

To find the direction of the third piece, we need to consider the angles provided in the problem.

Given:
m1 = 2.80 kg, v1 = 10.1 m/s, θ1 = 32.0° (above the horizontal)
m2 = 1.80 kg, v2 = 8.30 m/s, θ2 = 28.0° (below the horizontal)

To determine the direction of the third piece, we can use vector addition. We can break down the velocities of the second and third pieces into their horizontal and vertical components.

For the second piece (m2):
vx2 = v2 * cos(θ2)
vy2 = v2 * sin(θ2)

For the third piece:
vx3 = -v3 (since it travels backward)
vy3 = 0 (since it does not have an upward or downward component)

Now, we can add the horizontal and vertical components separately to determine the resultant velocity of the third piece.

vx_total = vx2 + vx3
vy_total = vy2 + vy3 = vy2 + 0 = vy2

Next, we can calculate the magnitude of the resultant velocity using the Pythagorean theorem:

v_total = √(vx_total^2 + vy_total^2)

Substituting the given values, we have:

v_total = √((-v3 + vx2)^2 + vy2^2)

Now, we have already calculated that the magnitude of the resultant velocity (v_total) is 48.738 m/s. We can use this information in the above equation to solve for the unknown velocity, v3.

48.738 = √((-v3 + vx2)^2 + vy2^2)

Now, we can square both sides of the equation to eliminate the square root:

48.738^2 = (-v3 + vx2)^2 + vy2^2

Expanding the equation:

(48.738^2) = (v3^2 - 2v3vx2 + vx2^2) + vy2^2

Rearranging the terms:

v3^2 - 2v3vx2 + vx2^2 + vy2^2 - (48.738^2) = 0

Now, we have a quadratic equation in terms of v3. By solving this quadratic equation, we can find the value of v3, which will give us the direction of the third piece.

However, before proceeding with the quadratic equation, please double-check your calculations and units to ensure accuracy.