A 0.30 kg puck, initially at rest on a frictionless horizontal surface, is struck by a 0.20 kg puck that is initially moving along the x axis with a velocity of 2.8 m/s. After the collision, the 0.20 kg puck has a speed of 0.5 m/s at an angle of θ = 53° to the positive x axis.

(a) Determine the velocity of the 0.30 kg puck after the collision.
(b) Find the fraction of kinetic energy lost in the collision.

(a) Well, well, well. Looks like we have a little collision party going on here. Let's get down to business.

We can start by using the principle of conservation of momentum. Before the collision, the total momentum is zero since the 0.30 kg puck is at rest. After the collision, the total momentum should still be zero. So, we can write the equation:

(m1)(v1) + (m2)(v2) = 0

where m1 and m2 are the masses of the pucks and v1 and v2 are their respective velocities.

Now, we plug in the given numbers. We have:

(0.30 kg)(v1) + (0.20 kg)(2.8 m/s) = 0

Solving for v1, we get:

v1 = -(0.20 kg)(2.8 m/s) / (0.30 kg)

Go ahead and crunch those numbers to find the velocity of the 0.30 kg puck after the collision. I'll wait, I'm a patient bot.

(b) Now onto the fraction of kinetic energy lost. The initial kinetic energy is given by:

KE_initial = (1/2)(m1)(v1_initial)^2 + (1/2)(m2)(v2_initial)^2

where v1_initial is the initial velocity of the 0.30 kg puck and v2_initial is the initial velocity of the 0.20 kg puck.

The final kinetic energy is given by:

KE_final = (1/2)(m1)(v1_final)^2 + (1/2)(m2)(v2_final)^2

where v1_final is the final velocity of the 0.30 kg puck and v2_final is the final velocity of the 0.20 kg puck.

The fraction of kinetic energy lost is then:

(KE_initial - KE_final) / KE_initial

So, you'll need to calculate the initial and final kinetic energies and then plug them into the formula. Get your calculator ready and start crunching those numbers. I'll be here to add a touch of humor if you need it!

To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.

(a) To determine the velocity of the 0.30 kg puck after the collision, we need to find the final velocities of both pucks. Let's denote the initial velocity of the 0.30 kg puck as v1_i, the initial velocity of the 0.20 kg puck as v2_i, the final velocity of the 0.30 kg puck as v1_f, and the final velocity of the 0.20 kg puck as v2_f.

According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision:

m1 * v1_i + m2 * v2_i = m1 * v1_f + m2 * v2_f

Substituting the given values:
0.30 kg * 0 + 0.20 kg * 2.8 m/s = 0.30 kg * v1_f + 0.20 kg * cos(θ) * v2_f

Since the 0.30 kg puck is initially at rest (v1_i = 0), the equation simplifies to:

0.20 kg * 2.8 m/s = 0.30 kg * v1_f + 0.20 kg * cos(θ) * v2_f

Now, let's solve for v1_f by isolating it in the equation:

0.20 kg * 2.8 m/s - 0.20 kg * cos(θ) * v2_f = 0.30 kg * v1_f

0.56 kg m/s - 0.20 kg * cos(53°) * v2_f = 0.30 kg * v1_f
0.56 kg m/s - 0.20 kg * (0.6) * v2_f = 0.30 kg * v1_f
0.56 kg m/s - 0.12 kg * v2_f = 0.30 kg * v1_f

Now, we need to calculate the final velocity of the 0.20 kg puck, v2_f, using the given speed of 0.5 m/s and angle θ = 53°. We can use the following formula:

v2_f = speed * cos(θ)

Substituting the values:
v2_f = 0.5 m/s * cos(53°)

v2_f ≈ 0.5 m/s * 0.6
v2_f ≈ 0.3 m/s

Now, we can substitute this value of v2_f back into the equation to find v1_f:

0.56 kg m/s - 0.12 kg * 0.3 m/s = 0.30 kg * v1_f

0.56 kg m/s - 0.036 kg m/s = 0.30 kg * v1_f

0.524 kg m/s = 0.30 kg * v1_f

v1_f = 0.524 kg m/s / 0.30 kg

v1_f ≈ 1.75 m/s

Therefore, the velocity of the 0.30 kg puck after the collision is approximately 1.75 m/s.

(b) To find the fraction of kinetic energy lost in the collision, we can use the formula:

Fraction of kinetic energy lost = (Initial kinetic energy - Final kinetic energy) / Initial kinetic energy

First, let's calculate the initial kinetic energy of the system:

Initial kinetic energy = 1/2 * m1 * v1_i^2 + 1/2 * m2 * v2_i^2

Substituting the given values:
Initial kinetic energy = 1/2 * 0.30 kg * 0^2 + 1/2 * 0.20 kg * 2.8 m/s^2

Initial kinetic energy = 0 + 1/2 * 0.20 kg * 2.8 m/s^2

Initial kinetic energy = 1/2 * 0.20 kg * 2.8 m/s^2

Initial kinetic energy = 0.28 J

Next, let's calculate the final kinetic energy of the system:

Final kinetic energy = 1/2 * m1 * v1_f^2 + 1/2 * m2 * v2_f^2

Substituting the calculated values:
Final kinetic energy = 1/2 * 0.30 kg * (1.75 m/s)^2 + 1/2 * 0.20 kg * (0.5 m/s)^2

Final kinetic energy = 1/2 * 0.30 kg * 3.0625 m^2/s^2 + 1/2 * 0.20 kg * 0.25 m^2/s^2

Final kinetic energy ≈ 0.4584375 J + 0.025 J

Final kinetic energy ≈ 0.4834375 J

Now let's calculate the fraction of kinetic energy lost:

Fraction of kinetic energy lost = (0.28 J - 0.4834375 J) / 0.28 J

Fraction of kinetic energy lost ≈ -0.2034375 J / 0.28 J

Fraction of kinetic energy lost ≈ -0.7265

Therefore, the fraction of kinetic energy lost in the collision is approximately 0.7265.

To determine the velocity of the 0.30 kg puck after the collision (part a), we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision. Mathematically, we can write the equation as:

m1 * v1_initial + m2 * v2_initial = m1 * v1_final + m2 * v2_final

where:
m1 = mass of the 0.30 kg puck
v1_initial = initial velocity of the 0.30 kg puck (which is at rest)
m2 = mass of the 0.20 kg puck
v2_initial = initial velocity of the 0.20 kg puck (2.8 m/s along the x-axis)
v1_final = final velocity of the 0.30 kg puck (which we need to find)
v2_final = final velocity of the 0.20 kg puck (0.5 m/s at angle θ = 53° to the positive x-axis)

Plugging in the given values, the equation becomes:

0.30 kg * 0 m/s + 0.20 kg * 2.8 m/s = 0.30 kg * v1_final + 0.20 kg * (0.5 m/s * cos θ + 0.5 m/s * sin θ)

Now, let's solve for v1_final by rearranging the equation:

v1_final = (0.20 kg * 2.8 m/s - 0.20 kg * (0.5 m/s * cos θ + 0.5 m/s * sin θ)) / 0.30 kg

Substituting the value of θ = 53° and evaluating the equation yields the answer for part a.

To find the fraction of kinetic energy lost in the collision (part b), we can calculate the initial kinetic energy and final kinetic energy and compare the two.

The initial kinetic energy is given by:

KE_initial = 0.5 * m1 * (v1_initial)^2 + 0.5 * m2 * (v2_initial)^2

Substituting the given values, we can calculate KE_initial.

The final kinetic energy is given by:

KE_final = 0.5 * m1 * (v1_final)^2 + 0.5 * m2 * (v2_final)^2

Substituting the values of v1_final and v2_final obtained from part a, we can calculate KE_final.

The fraction of kinetic energy lost is then given by:

Fraction of kinetic energy lost = (KE_initial - KE_final) / KE_initial

Substituting the values of KE_initial and KE_final, we can calculate the answer for part b.

(a) Use the same method i suggested here:

http://www.jiskha.com/display.cgi?id=1319256261

Part (b) will be easy once you do part (a).
Just compare initial kinetic energy with final total kinetic energy of both pucks.