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
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
Let's first define our variables:
- v1 = initial velocity of the cue ball
- v2 = initial velocity of the target ball
- θ1 = angle between the velocity of the cue ball after the collision and its original direction of travel
- θ2 = angle between the velocity of the target ball after the collision and its original direction of travel
- v1' = final velocity of the cue ball after the collision
- v2' = final velocity of the target ball after the collision
(a) Find the angle between the velocity vectors of the two balls after the collision:
Since we know the angle between the velocity vectors of the cue ball before and after the collision is 30°, we can say θ1 = 30°. To find θ2, we need to use the concept of conservation of momentum.
Conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it.
Initially, the total momentum of the system is zero because the target ball is at rest. After the collision, the total momentum of the system remains zero.
Mathematically, we can write:
m1 * v1 + m2 * v2 = m1 * v1' + m2 * v2'
Since both balls have equal mass, we can simplify the equation to:
v1 + v2 = v1' + v2' -------- (Equation 1)
Now, let's focus on the angles.
The angle between the velocity vector of the target ball before the collision and its final velocity vector after the collision is θ2.
Since we know θ1 = 30°, we can use the relationship between angles during elastic collisions, which states that the angle of incidence is equal to the angle of reflection.
θ1 = θ2
Therefore, θ2 = 30°.
(b) Find the speed of each ball after the collision:
To find the speeds of the balls after the collision, we can use the principle of conservation of kinetic energy.
Conservation of kinetic energy states that the total kinetic energy of a closed system remains constant if no external forces act on it.
The kinetic energy of an object is given by the formula:
KE = 0.5 * m * v^2
Initially, the system has kinetic energy only due to the movement of the cue ball. After the collision, the cue ball and the target ball will have their own kinetic energies.
So we can write:
0.5 * m1 * (v1^2) = 0.5 * m1 * (v1'^2) + 0.5 * m2 * (v2'^2) -------- (Equation 2)
Substituting Equation 1 into Equation 2, we get:
0.5 * m1 * (v1^2) = 0.5 * m1 * ((v1' + v2')^2) + 0.5 * m2 * (v2'^2)
Simplifying further, we have:
v1^2 = v1'^2 + 2 * v1' * v2' + v2'^2 ----- (Equation 3)
Now, we have two equations (Equation 1 and Equation 3) with two unknowns (v1' and v2'). By solving these equations simultaneously, we can find the values of v1' and v2', which will give us the speeds of the cue ball and the target ball after the collision. This can be solved using algebraic or graphical methods.