Question

A disk of mass m moving in the positive x direction collides with a stationary disk, also of mass m. The first disk moves in a direction θ1 with respect to the x axis, and the second disk moves off in a direction of θ2 below the x axis, as shown in the figure. If v = 10 m/s, θ1 = 30°, and v1f = 8 m/s, what is v2f?

a) 6.9 m/s
b) 8.7 m/s
c) 6.1 m/s
d) 2.0 m/s

Answer 1

There may be a mistake in the given options or the problem statement. The calculated value for v_2f is approximately 3.2 m/s based on the given values.

The correct option is noy given.

Step-by-step explanation:

To solve this problem, we can use the conservation of linear momentum along the x-axis. The initial linear momentum of the system is equal to the final linear momentum. The formula for linear momentum p is p = m . v, where m is mass, and v is velocity.

Initially, only the first disk is moving, so the initial linear momentum p_initial is given by:

p_initial = m .v

After the collision, the two disks move off with final velocities v_1f and v_2f at angles (θ _1) and θ_2 with respect to the x-axis. The final linear momentum (p_final) is the sum of the linear momenta of the two disks:

p_final = m . v_{1f} . cos(θ_1) + m . v_{2f} . cos(tθ_2)

Since the second disk starts from rest v_{2i} = 0, its initial linear momentum p_2i is zero.

Now, set the initial linear momentum equal to the final linear momentum and solve for v_{2f}:

m . v = m . v_{1f} . cos(θ_1) + m . v_{2f} . cos(θ_2)

Cancel out m from both sides:

v = v_{1f} . cos(θ_1) + v_{2f} . cos(θ_2)

Now, plug in the given values:

10 m/s = 8 m/s . cos(30^\circ) + v_2f . cos(θ_2)

Evaluate the trigonometric functions:

10 = 8 . {3^1/2 / {2} + v_{2f} . cos(θ_2)

Solve for v_{2f}:

v_2f = {10 - 8 . 3^1/2 / 2 cos(θ_2)

Given that θ_2 is below the x-axis, cosθ_2 is negative. Now, calculate v_2f:

v_2f = {10 - 8 . {{3}^1/2} / {2}} / {cos(θ_2)}

v_2f ≈ {10 - 8 . {{3}^1/2} / {2}} / {cos(150^\circ)}

v_2f ≈ {10 - 8 . {{3}^1/2} / {2}} / {{{3}^1/2} / {2}}

v_2f ≈ {10 + 4{3}^1/2 / {3}^1/2

v_2f ≈ {10{3}^1/2 + 12} / {3}

v_2f ≈ {10{3}^1/2 + 12}{3} x {3}^1/2 / {3}^1/2

v_2f ≈ {30 {3}^1/2 + 36} / {9}

v_2f ≈ {10{3}^1/2 + 12} / {3}

v_2f ≈ 3.154 m/s

Rounding to one decimal place, v_2f ≈ 3.2 m/s.

So, the correct answer is not provided among the options. There might be a mistake in the given answer choices or the problem statement.