A cue ball traveling at 4.0 m/s makes a glancing, elastic collision with a target ball of equal mass that is initially at rest. The cue ball is deflected so that it makes an angle of 30° with its original direction of travel.
(a) Find the angle between the velocity vectors of the two balls after the collision.
°
(b) Find the speed of each ball after the collision.
cue ball m/s
target ball m/s
(a) Well, we've got a cue ball and a target ball going head-to-head in a hilarious collision. Now, after this comedic encounter, we are asked to find the angle between the velocity vectors of the two balls.
Since there is an angle involved, it seems like a job for some trigonometry! The initial direction of the cue ball is at an angle of 30° from its final direction. So, the angle between the velocity vectors of the two balls after the collision must be 30°.
(b) Now, let's shift gears and find the speeds of each ball after the collision.
Since it's an elastic collision, we know that both momentum and kinetic energy are conserved. Since the masses are equal, it means that the speed of the cue ball after the collision would be the same as its initial speed, which is 4.0 m/s.
For the target ball, we can use the conservation of momentum to find its speed after the collision. Since it was initially at rest, its initial momentum is zero. The final momentum must also be zero to satisfy the conservation of momentum. Therefore, the speed of the target ball after the collision would be zero.
So, after this comedic episode, the cue ball will be cruising along at 4.0 m/s, while the target ball will be left standing still, perhaps contemplating the meaning of life.
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy for an elastic collision.
Let's first calculate the angle between the velocity vectors of the two balls after the collision.
(a) The angle between the velocity vectors of the two balls after the collision can be found using the Law of Sines. The equation is as follows:
sin(theta) = (m1 * v1 * sin(phi1)) / (m2 * v2 * sin(phi2))
where:
- theta is the angle between the velocity vectors after the collision,
- m1 and m2 are the masses of the cue ball and target ball (which are equal),
- v1 and v2 are the velocities of the cue ball and target ball after the collision,
- phi1 is the angle between the initial velocity vector of the cue ball and the collision direction,
- phi2 is the angle between the initial velocity vector of the target ball and the collision direction.
In this case, v1 (velocity of the cue ball after the collision) is not given. However, we know that the cue ball is deflected so that it makes an angle of 30° with its original direction of travel. This means that phi1 is 30°.
Since the target ball is initially at rest, its initial velocity vector is also the collision direction. Hence, phi2 is 0°.
Plugging these values into the equation, we get:
sin(theta) = (m1 * v1 * sin(phi1)) / (m2 * v2 * sin(phi2))
sin(theta) = (m1 * v1 * sin(30°)) / (m2 * v2 * sin(0°))
sin(30°) = (v1 * sin(30°)) / v2
0.5 = (v1 * 0.5) / v2
v1 = v2
So, the angle between the velocity vectors of the two balls after the collision is 0°.
(b) Since the angles are equal and the masses are equal, the velocities of the two balls after the collision are also equal.
The speed of each ball after the collision can be calculated using the conservation of kinetic energy. The equation is as follows:
(0.5 * m1 * (v1)^2) + (0.5 * m2 * (v2)^2) = (0.5 * m1 * (v1f)^2) + (0.5 * m2 * (v2f)^2)
Since v1 = v2, and both balls have equal masses,
(0.5 * v1^2) + (0.5 * v1^2) = (0.5 * v1f^2) + (0.5 * v1f^2)
v1^2 = v1f^2
v1 = v1f
So, the speed of each ball after the collision is equal to their initial speed, which is 4.0 m/s.
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
(a) Using conservation of momentum, we know that the total momentum before the collision is equal to the total momentum after the collision. The momentum of an object is defined as the product of its mass and velocity. Since the mass of the cue ball and the target ball are equal, we can express this as:
Initial momentum before collision = Final momentum after collision
The initial momentum before the collision is simply the momentum of the cue ball, which we can calculate as the product of its mass and velocity:
Initial momentum = mass of cue ball * velocity of cue ball
Similarly, the final momentum after the collision will be the sum of momentum of the cue ball and the momentum of the target ball:
Final momentum = momentum of cue ball + momentum of target ball
Since the target ball is initially at rest, its momentum is zero. So we can write:
mass of cue ball * velocity of cue ball = momentum of cue ball + 0
Now, let's solve for the angle between the velocity vectors of the two balls after the collision.
To find the angle, we can use the law of cosines. The law of cosines states that in a triangle with sides a, b, and c, and the angle opposite to side c (let's call it angle C), we have:
c^2 = a^2 + b^2 - 2ab * cos(C)
In our case, the sides a and b are the magnitudes of the velocities of the cue ball and target ball after the collision, and side c is the magnitude of their relative velocity. Since we are given the angle between the velocity vectors of the cue ball after the collision and its original direction of travel (30°), we can use the law of cosines to find the magnitude of the relative velocity, and then calculate the angle C.
(b) Using the law of conservation of kinetic energy, we know that the total kinetic energy before the collision is equal to the total kinetic energy after the collision. The kinetic energy of an object is defined as one-half times its mass times the square of its velocity. We can express this as:
Initial kinetic energy before collision = Final kinetic energy after collision
The initial kinetic energy before the collision is simply the kinetic energy of the cue ball, which we can calculate using its mass and velocity:
Initial kinetic energy = (1/2) * mass of cue ball * (velocity of cue ball)^2
Similarly, the final kinetic energy after the collision will be the sum of the kinetic energy of the cue ball and the kinetic energy of the target ball:
Final kinetic energy = (1/2) * (mass of cue ball) * (velocity of cue ball)^2 + (1/2) * (mass of target ball) * (velocity of target ball)^2
Since the mass of the target ball is equal to the mass of the cue ball, we can simplify this expression to:
Final kinetic energy = (1/2) * (mass of cue ball) * [(velocity of cue ball)^2 + (velocity of target ball)^2]
Now, let's solve for the velocities of the cue ball and target ball after the collision.
By equating the final kinetic energy to the initial kinetic energy, we can solve for the unknown velocities.