While playing a game of billiards, your 0.50 kg cue ball, travelling at 1.9 m/s, glances off a stationary 0.30 kg billiard ball so that the billiard ball moves off at 1.3 m/s at an angle of 32º clockwise from the cue ball’s original path. What is the final speed of the cue ball?
my bad i messed up the real answer i got is v= 1.3 m/s
i got .41 m/s
To solve this problem, we can apply the law of conservation of momentum. According to this law, the total momentum before the collision should be equal to the total momentum after the collision.
We can start by calculating the momentum of the cue ball before the collision. Momentum (p) is given by the equation:
p = mass × velocity
Given that the mass of the cue ball (m1) is 0.50 kg and its velocity (v1) is 1.9 m/s, we can calculate its momentum:
p1 = m1 × v1
= 0.50 kg × 1.9 m/s
= 0.95 kg·m/s
Next, we can calculate the momentum of the billiard ball after the collision. We know that it moves off at a velocity of 1.3 m/s at an angle of 32º clockwise from the cue ball's original path. We can break down this velocity into horizontal and vertical components.
The horizontal component of the billiard ball's velocity (v2x) can be found using trigonometry:
v2x = v2 × cos(angle)
= 1.3 m/s × cos(32º)
≈ 1.10 m/s
The vertical component of the billiard ball's velocity (v2y) can also be found using trigonometry:
v2y = v2 × sin(angle)
= 1.3 m/s × sin(32º)
≈ 0.69 m/s
Now, we can calculate the momentum of the billiard ball after the collision:
p2 = mass × velocity
= m2 × (v2x + v2y)
Given that the mass of the billiard ball (m2) is 0.30 kg, we can substitute the values and calculate its momentum.
p2 = 0.30 kg × (1.10 m/s + 0.69 m/s)
≈ 0.57 kg·m/s
According to the conservation of momentum, the total momentum before the collision (p1) should be equal to the total momentum after the collision (p2).
p1 = p2
0.95 kg·m/s = 0.57 kg·m/s
To find the final speed of the cue ball, we need to calculate its final velocity (v1f). We can rearrange the momentum equation to solve for v1f:
p1 = m1 × v1f
Dividing both sides of the equation by m1:
v1f = p1 / m1
= 0.95 kg·m/s / 0.50 kg
= 1.9 m/s
Therefore, the final speed of the cue ball is 1.9 m/s.