Two pool balls, each of mass 0.2kg, collide as shown in the figure above. Before the collision, the black ball’s velocity makes an angle of 30∘ with the horizontal line. After the collision, the white ball’s velocity makes an angle of 60∘ with the vertical line. What is the black ball’s speed after the collision?
If needed, sin(30∘)=cos(60∘)=12,sin(60∘)=cos(30∘)=3√2,tanθ=sinθcosθ
sin(90∘+θ)=cosθ,cos(90∘+θ)=−sinθ,g=10ms2
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Speed =- unanswered m/s
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To find the black ball's speed after the collision, we can use the principle of conservation of momentum.
1. First, let's calculate the initial horizontal and vertical components of the black ball's velocity.
Given:
Mass of the black ball (m1) = 0.2 kg
Angle with the horizontal before collision = 30 degrees
The initial horizontal component of velocity (V1x) can be calculated as:
V1x = velocity * cos(angle)
V1x =V1 * cos(30°)
V1x = V1 * ( √3 / 2)
The initial vertical component of velocity (V1y) can be calculated as:
V1y = velocity * sin(angle)
V1y = V1 * sin(30°)
V1y = V1 * (1/2)
Therefore, the initial velocity components are:
V1x = V1 * ( √3 / 2)
V1y = V1 * (1/2)
2. Next, let's calculate the final horizontal and vertical components of the white ball's velocity.
Given:
Angle with the vertical after collision = 60 degrees
The final horizontal component of velocity (V2x) can be calculated as:
V2x = velocity * sin(angle)
V2x = V2 * sin(60°)
V2x = V2 * ( √3 / 2)
The final vertical component of velocity (V2y) can be calculated as:
V2y = velocity * cos(angle)
V2y = V2 * cos(60°)
V2y = V2 * (1/2)
Therefore, the final velocity components are:
V2x = V2 * ( √3 / 2)
V2y = V2 * (1/2)
3. Now, let's apply the principle of conservation of momentum.
The momentum in the x-direction before the collision is equal to the momentum in the x-direction after the collision. Similarly, the momentum in the y-direction before the collision is equal to the momentum in the y-direction after the collision.
In the x-direction:
Initial momentum = m1 * V1x
Final momentum = m2 * V2x
(0.2 kg) * (V1 * ( √3 / 2)) = (0.2 kg) * (V2 * ( √3 / 2))
In the y-direction:
Initial momentum = m1 * V1y
Final momentum = m2 * V2y
(0.2 kg) * (V1 * (1/2)) = (0.2 kg) * (V2 * (1/2))
Since the masses are the same (0.2 kg), we can remove them from the equations.
V1 * √3 = V2 * √3 --- (Equation 1)
V1 * 1/2 = V2 * 1/2 --- (Equation 2)
4. We can solve equations 1 and 2 simultaneously to find the ratio of V1 to V2.
Dividing equation 1 by equation 2, we get:
(V1 * √3) / (V1 * 1/2) = (V2 * √3) / (V2 * 1/2)
2√3 = 2√3
Therefore, the ratio of V1 to V2 is 1:1, or V1 = V2.
5. Since the black ball's speed after the collision is equal to V2, we can conclude that the black ball's speed after the collision is equal to its initial speed, which is V1.
Therefore, the black ball's speed after the collision is V1.
Speed = V1
Therefore, the black ball's speed after the collision is V1.
To find the black ball's speed after the collision, we can use the principle of conservation of linear momentum and apply it to the horizontal and vertical components of the velocities.
Let's break down the problem and solve it step by step:
1. Start by analyzing the horizontal component of the velocities:
- Before the collision, the black ball's velocity makes an angle of 30∘ with the horizontal line. We can use this information to find the initial horizontal velocity of the black ball. Let's call it Vx1.
- Given that sin(30∘) = 1/2, we can write Vx1 = V1 * cos(30∘), where V1 represents the magnitude of the initial velocity of the black ball.
2. Next, analyze the vertical component of the velocities:
- After the collision, the white ball's velocity makes an angle of 60∘ with the vertical line. We can use this information to find the final vertical velocity of the black ball. Let's call it Vy2.
- Given that cos(60∘) = 1/2, we can write Vy2 = V2 * sin(60∘), where V2 represents the magnitude of the final velocity of the black ball.
3. Apply the conservation of linear momentum in the vertical direction:
- The total vertical momentum before the collision is zero since both the black and white ball have zero vertical velocity initially.
- After the collision, the black ball's horizontal velocity remains unchanged, so the total vertical momentum after the collision is also zero.
- From this, we can conclude that Vy2 = 0.
4. Calculate the speed of the black ball after the collision:
- The velocity of the black ball after the collision can be found using the Pythagorean theorem because we now know the horizontal and vertical components of its velocity.
- We can write V2 = √(Vx2^2 + Vy2^2), where Vx2 represents the magnitude of the horizontal velocity of the black ball after the collision.
- Since Vy2 is zero, V2 = √(Vx2^2 + 0^2) = Vx2.
Therefore, we need to find Vx1, the initial horizontal velocity of the black ball, and then we can set V2 = Vx2 = Vx1 to find the black ball's speed after the collision.
Please provide the value of the initial horizontal velocity (Vx1) to proceed with the calculation.