A 9.1 g ball is hit into a 98 g block of clay at rest on a level surface. After impact, the block slides 8.0 m before coming to rest. If the coefficient of friction is 0.60, determine the speed of the ball before impact.

initial ball speed = vi

momentum before = .0091 vi
momentum after = .0091 vi = (.107)vf
so
vf = .085 vi

work done by friction = initial Ke of clay with ball

.6 (.107)(9.81)(8) = (1/2)(.107)(vf^2)
so vf = 8.86 m/s
vi = 8.86/.085 = 104 m/s
check my arithmetic !

To solve this problem, we need to use the concept of conservation of momentum and the work-energy theorem.

Step 1: Determine the initial momentum of the system before impact.
The momentum (p) is calculated by multiplying the mass (m) of an object by its velocity (v). Since the block of clay is initially at rest, the initial momentum of the system is equal to the momentum of the ball: p_initial = m_ball * v_ball.

Step 2: Determine the final momentum of the system after impact.
The final momentum of the system is zero since both the ball and the block come to rest after the impact. Hence, p_final = 0.

Step 3: Apply the conservation of momentum.
According to the conservation of momentum, the initial momentum is equal to the final momentum. Therefore, p_initial = p_final. We can write this equation as m_ball * v_ball = 0.

Step 4: Solve for the velocity of the ball.
Rearranging the equation, we find v_ball = 0 / m_ball. Since the mass of ball (m_ball) is positive, the velocity of the ball (v_ball) must be zero. However, this does not represent the velocity before impact. So, we need to consider the work done by the friction force.

Step 5: Calculate the work done by the friction force.
The work done by the friction force (W_friction) is given by the product of the friction force (f_friction) and the distance over which it acts (d): W_friction = f_friction * d.

Step 6: Calculate the friction force.
The friction force can be calculated using the equation f_friction = coefficient of friction * normal force. The normal force is equal to the weight of the block, which is given by the product of its mass (m_block) and the acceleration due to gravity (g). Hence, f_friction = μ * m_block * g, where the coefficient of friction (μ) is 0.60, the mass of the block (m_block) is 98 g = 0.098 kg, and the acceleration due to gravity (g) is approximately 9.8 m/s².

Step 7: Calculate the work done by friction.
Substituting the values into the equation, we have W_friction = 0.60 * 0.098 kg * 9.8 m/s² * 8.0 m.

Step 8: Calculate the work done by the friction force.
Using the work-energy theorem, the work done by the friction force is equal to the initial kinetic energy of the ball before impact. Hence, W_friction = (1/2) * m_ball * v_ball².

Step 9: Solve for the velocity of the ball before impact.
Rearranging the equation, we find v_ball² = 2 * W_friction / m_ball. Substituting the values, v_ball² = (2 * (0.60 * 0.098 kg * 9.8 m/s² * 8.0 m)) / 9.1 g.

Step 10: Calculate the velocity of the ball before impact.
Taking the square root of both sides, we find v_ball = √(2 * W_friction / m_ball). Evaluating the expression, v_ball = √((2 * (0.60 * 0.098 kg * 9.8 m/s² * 8.0 m)) / 9.1 g).

To determine the speed of the ball before impact, we need to apply the principles of conservation of momentum and kinetic energy.

Let's break down the problem:

1. Conservation of momentum: According to Newton's third law of motion, the total momentum before impact is equal to the total momentum after impact as long as there are no external forces acting. We can express the momentum as the product of mass and velocity.

Total momentum before impact = Total momentum after impact

2. Kinetic energy: The work done by friction, which causes the block to stop, can be obtained by multiplying the coefficient of friction by the normal force and the distance over which the block slides.

Work done by friction = μ * (mass of the block) * (acceleration due to gravity) * (distance)

Using these concepts, let's calculate the speed of the ball before impact step by step:

Step 1: Find the momentum of the block after impact.
The mass of the block is given as 98 g, which is 0.098 kg.
The initial velocity of the block is 0 m/s since it is at rest.
The final velocity of the block is also 0 m/s since it comes to rest.
Therefore, the momentum of the block after impact is:
Momentum of the block after impact = (mass of the block) * (final velocity of the block) = 0.098 kg * 0 m/s = 0 kg·m/s.

Step 2: Find the work done by friction.
The coefficient of friction is given as 0.60.
The distance over which the block slides is given as 8.0 m.
The normal force can be calculated using the equation:
Normal force = (mass of the block) * (acceleration due to gravity)
Normal force = 0.098 kg * 9.8 m/s² = 0.9604 N
Therefore, the work done by friction is:
Work done by friction = (coefficient of friction) * (normal force) * (distance) = 0.60 * 0.9604 N * 8.0 m = 4.60512 J

Step 3: Apply conservation of momentum.
The momentum of the ball after impact can be written as:
Momentum of the ball after impact = (mass of the ball) * (velocity of the ball after impact)

The momentum of the ball before impact can be expressed as:
Momentum of the ball before impact = (mass of the ball) * (velocity of the ball before impact)

Since the total momentum before impact is equal to the total momentum after impact, we have:
(mass of the ball) * (velocity of the ball before impact) = (mass of the ball) * (velocity of the ball after impact) + Momentum of the block after impact

Substituting the given values, we get:
9.1 g = 0.0091 kg (mass of the ball)
0 kg·m/s (velocity of the ball after impact)

Solving the equation, we find:
(0.0091 kg) * (velocity of the ball before impact) = 0.0091 kg * 0 kg·m/s + 0 kg·m/s

Therefore, the velocity of the ball before impact is:
Velocity of the ball before impact = 0 kg·m/s / 0.0091 kg ≈ 0 m/s

So, the speed of the ball before impact is approximately 0 m/s.