A bowling ball of mass M and radius R is thrown along a level surface so that initially ( t= 0) it slides with a linear speed but does not rotate, see the figure. As it slides, it begins to spin, and eventually rolls without slipping.

How far has the ball moved down the lane when it starts rolling without slipping?

What are its final linear speed?

What are its final rotational speed?

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To find the distance the ball has moved down the lane when it starts rolling without slipping, we need to consider the situation when the ball transitions from sliding to rolling without slipping.

Let's use the following variables:
- m: mass of the ball
- r: radius of the ball
- v_initial: initial linear speed of the ball
- d: distance the ball has moved down the lane when it starts rolling without slipping

When the ball slides, there is kinetic friction between the ball and the surface, which provides a torque on the ball. This torque causes the ball to start rotating. At some point, the torque from friction will be equal to the torque exerted by the rotational inertia of the ball.

The torque from friction is given by:
τ_friction = μ_s * m * g * r,

where μ_s is the coefficient of sliding friction, m is the mass of the ball, g is the acceleration due to gravity, and r is the radius of the ball.

The torque exerted by the rotational inertia of the ball is given by:
τ_rotational = I * α,

where I is the moment of inertia of the ball and α is the angular acceleration.

In this case, since the ball is initially not rotating, the moment of inertia of the ball is given by:
I = 2/5 * m * r^2,

which is for a solid sphere.

The angular acceleration α is related to the linear acceleration a by the radius of the ball:
α = a / r.

The force causing linear acceleration a is the gravitational force acting on the ball, which can be calculated as:
F_gravity = m * g.

The linear acceleration a can be obtained using the second law of motion:
F_gravity - F_friction = m * a,

where F_friction is the force of friction acting on the ball. The force of friction can be calculated as:
F_friction = μ_k * m * g,

where μ_k is the coefficient of kinetic friction.

By substituting the above equations, we have:
μ_k * m * g = m * a.

Solving for a, we get:
a = μ_k * g.

Now that we have the linear acceleration, we can find the time it takes for the ball to transition from sliding to rolling without slipping. During this time, the ball will cover a distance d down the lane. We can use the equation of motion:
d = v_initial * t + 1/2 * a * t^2.

At this point, when the ball has traveled a distance d, the linear velocity of the ball will be equal to the angular velocity times the radius of the ball:
v_linear = ω * r,

where ω is the angular velocity of the ball.

To find the final linear speed, we need to determine the linear speed when the ball is rolling without slipping. When the ball has transitioned to rolling without slipping, the linear speed is equal to the angular speed times the radius of the ball:
v_linear_final = ω_final * r.

To find the final rotational speed, we use the concept of conservation of angular momentum. Before the ball starts rolling without slipping, its angular momentum is zero since it is not rotating. After the ball starts rolling without slipping, its angular momentum is given by the equation:
L_final = I * ω_final.

Once we have the final angular momentum, we can calculate the final rotational speed using the equation:
L_final = I * ω_final,

where ω_final is the final rotational speed.

By solving these equations, you can find the distance the ball has moved down the lane when it starts rolling without slipping, its final linear speed, and its final rotational speed.