A billiard ball of mass 0.16 kg has a speed of 1.80 m/s and collides with the side of the billiard table at an angle of 34.6°. For this collision, the coefficient of restitution is 0.841. What is the angle relative to the side (in degrees) at which the ball moves away from the collision?
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To find the angle relative to the side at which the ball moves away from the collision, we can use the concept of conservation of momentum and the coefficient of restitution.
The coefficient of restitution (e) is a measure of the "bounciness" of a collision. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach.
Given:
Mass of the billiard ball (m) = 0.16 kg
Initial speed of the ball (v) = 1.80 m/s
Angle of collision with the side (θ) = 34.6°
Coefficient of restitution (e) = 0.841
Step 1: Resolve the initial velocity of the ball into horizontal and vertical components.
The horizontal component (v₀x) of the initial velocity is given by:
v₀x = v₀ * cos(θ)
The vertical component (v₀y) of the initial velocity is given by:
v₀y = v₀ * sin(θ)
Plugging in the values:
v₀x = 1.80 m/s * cos(34.6°)
v₀y = 1.80 m/s * sin(34.6°)
Step 2: Calculate the horizontal and vertical components of the final velocity.
Since there is no friction acting horizontally, the horizontal component of the final velocity remains the same after the collision.
v₁x = v₀x
To find the vertical component of the final velocity (v₁y), we use the formula:
v₁y = -e * v₀y
Plugging in the values:
v₁y = -0.841 * (1.80 m/s * sin(34.6°))
Step 3: Find the magnitude and direction of the final velocity.
The magnitude of the final velocity (v₁) is given by:
v₁ = sqrt(v₁x² + v₁y²)
The direction of the final velocity (θ₁) can be found using the inverse tangent function:
θ₁ = arctan(v₁y / v₁x)
Plugging in the values:
v₁ = sqrt((v₁x)² + (v₁y)²)
θ₁ = arctan(v₁y / v₁x)
Finally, the angle relative to the side at which the ball moves away from the collision is the complement of θ₁ with respect to the side. To calculate it, subtract θ₁ from 90°.
Step 4: Calculate the angle relative to the side.
Angle relative to the side = 90° - θ₁
Now you have all the steps to calculate the angle relative to the side at which the ball moves away from the collision. Simply plug in the values and solve for the angle.