A billiard ball of mass 0.55kg moves with a velocity of 12.5m/s towards a stationary billiard ball of the identical mass and strikes it with a glancing blow. The first billiard ball moves of at an angle of 29.7deg clockwise from its original direction, with a velocity of 9.56m/s. Determine whether the collision was elastic.
Ans: since it is 2D collision, i tired solving it before and after; vertical component and horizontal component, i keep getting my second angle as 54deg and velocity 7m/s, where as my answer should be 41deg and 6m/s.
Before
Vertical
0.155(12.5)
=1.9375
Horizontal = 0
After
Vertical
(0.155)(9.56)(cos 29.7) + (0.155)(V'sin Theta) = 1.9375
(0.155)(V'sin Theta) = 1.9375 - (0.155)(9.56)(cos 29.7) (eq 1)
Horizontal
(0.155)(9.56)(V' cos 29.7) -(0.155)(V' cos theta) = 0
(0.155)(V' cos theta) = -(0.155)(9.56)(V' cos 29.7) (eq 2)
Then you solve simultaneously, but i do not get the answer
We do not know if it is an elastic collision, but momentum is always conserved.
Your calculations seem to be based on the conservation of momentum in both directions, but you had a cos(θ) for both the "horizontal", or x-direction, and the "vertical", or y-direction.
Assuming the original direction of the first ball is along the y-axis, then the deviation of its course, θ, is -29.7°, and the new velocity, v1=9.56 m/s.
The stationary ball now moves with a velocity v2, and at an angle φ with the y-axis.
Equate momentum in the x-direction:
v1*sin(θ)+v2*sin(φ)=0
Equate momentum in the y-direction:
v1*cos(θ)+v2*cos(φ)=12.5
Solving the equations, I get v2=6.3 and φ=48°, different from your answer.
After that, you equate initial KE and final KE to determine if the collision was elastic.
How is it possible that mass which is 0.155kg is not taken into consideration?
You solve it as a 2D collision and thereafter you obtain the two equations, two variables and you solve you will get V'
Thereafter you find if it is elastic or not by equating the kinetic energy
To determine whether the collision between the billiard balls is elastic, we need to analyze the conservation of kinetic energy and momentum.
Step 1: Calculate the initial momentum before the collision:
Since both billiard balls have the same mass, their initial momentum is simply the mass of one ball multiplied by the velocity of the moving ball:
Initial momentum = (0.55 kg) * (12.5 m/s)
Step 2: Calculate the final momentum after the collision:
The final momentum can be split into horizontal and vertical components. Let's denote the horizontal component as Px' and the vertical component as Py'.
The vertical component of momentum (Py') can be calculated using the mass of the ball and its vertical velocity after the collision:
Py' = (0.55 kg) * (9.56 m/s * sin(29.7°))
The horizontal component of momentum (Px') can be calculated using the mass of the ball and its horizontal velocity after the collision:
Px' = (0.55 kg) * (9.56 m/s * cos(29.7°))
Step 3: Calculate the initial kinetic energy before the collision:
The initial kinetic energy is given by the formula: Initial kinetic energy = (1/2) * mass * (velocity)^2 for both billiard balls.
Step 4: Calculate the final kinetic energy after the collision:
The final kinetic energy is given by: Final kinetic energy = (1/2) * mass * (velocity)^2 for both billiard balls.
Step 5: Compare the initial and final kinetic energies:
If the initial kinetic energy is equal to the final kinetic energy, the collision is perfectly elastic. If they are not equal, the collision is not perfectly elastic.
Step 6: Calculate the angles and velocities after the collision:
Let's denote the angle of the first ball after the collision as Theta' and its velocity as V'.
Using the derived equations, we can solve for Theta' and V' by equating the expressions for momentum (Px' and Py') and solving simultaneously.
By following these steps, you should be able to determine whether the collision was elastic and calculate the correct values for the angles and velocities after the collision.
To determine whether the collision was elastic, we need to examine the conservation of kinetic energy and momentum before and after the collision.
Let's first analyze the momentum. The total momentum before the collision is given by:
Initial momentum = (mass of the first ball) * (velocity of the first ball)
= (0.55 kg) * (12.5 m/s)
The total momentum after the collision can be calculated by decomposing the velocities into their vertical and horizontal components. Thus, we have:
Vertical component of momentum after collision = (mass of the first ball) * (final velocity of the first ball) * cos(angle)
Horizontal component of momentum after collision = (mass of the first ball) * (final velocity of the first ball) * sin(angle)
Using the given values, we can write equations for momentum conservation in the vertical and horizontal directions:
Equation 1: (0.55 kg) * (9.56 m/s) * cos(29.7°) + (0.55 kg) * (final velocity of the first ball) * sin(final angle) = (0.55 kg) * (12.5 m/s)
Equation 2: (0.55 kg) * (final velocity of the first ball) * cos(final angle) = 0
Now, let's consider the kinetic energy. The initial kinetic energy can be calculated using the formula:
Initial kinetic energy = 0.5 * (mass of the first ball) * (velocity of the first ball)^2
The final kinetic energy can be obtained by calculating the kinetic energy of the first ball after the collision:
Final kinetic energy = 0.5 * (mass of the first ball) * (final velocity of the first ball)^2
To determine whether the collision is elastic, we compare the initial and final kinetic energies. If they are the same, the collision is elastic; otherwise, it is inelastic.
Now, using the equations for momentum conservation and the formulas for kinetic energy, you can solve for the final velocity and angle of the first ball after the collision. Substitute the values into the equations, solve them simultaneously, and check if you obtain the expected final velocity and angle. If not, review your calculations to ensure accuracy.