Question

In bocce, the object of the game is to get your balls (each with mass \(M=1.00 \mathrm{~kg}\) ) as close as possible to the small white ball (the pallina, mass \(m=0.045 \mathrm{~kg}\) ). Your first throw positioned your ball \(2.00 \mathrm{~m}\) to the left of the pallina. If your next throw has a speed of \(v=1.00 \mathrm{~m} / \mathrm{s}\) and the coefficient of kinetic friction is \(\mu_{\mathrm{k}}=0.20\), what are the final distances of your two balls from the pallina in each of the following cases? a) You throw your ball from the left, hitting your first ball. b) You throw your ball from the right, hitting the pallina.

Short answer

Answer: To find the final distances, first, calculate the final velocities of the balls and pallina using the conservation of momentum and Newton's second law equations. Next, use kinematic equations and the calculated accelerations and velocities to find the distances traveled by the balls and pallina. Finally, add or subtract these distances from their initial positions in each case to find the final distances of the balls from the pallina.

Step-by-step solution

Understand the situation and gather information

In this step, we need to understand the given problem and collect all the necessary information, which includes the mass of the balls, initial positions and velocities, and the coefficient of kinetic friction. Given data: - Mass of the first ball (M) = 1.00 kg - Mass of the pallina (m) = 0.045 kg - Initial position of your first ball = 2.00 m to the left of the pallina - The speed of your next throw (v) = 1.00 m/s - The coefficient of kinetic friction (μk) = 0.20

Apply conservation of momentum principle

Before the collision, the momentum of all the objects involved must equal the total momentum after the collision. Using the conservation of momentum principle, we can write the equation for each case: a) When you throw your ball from the left and hit the first ball: Initial momentum = Final momentum M * v + M * 0 = M * v1 + M * v2 b) When you throw your ball from the right and hit the pallina: Initial momentum = Final momentum m * 0 + M * v = m * vp + M * vr Where: - v1 = The final velocity of your first ball in case a - v2 = The final velocity of your second ball in case a - vp = The final velocity of the pallina in case b - vr = The final velocity of your second ball in case b

Calculate final velocities using Newton's second law

Using Newton's second law (F = ma), we can calculate the final velocities of the balls and pallina, considering the friction forces acting on them: a) Velocity of first ball in case a: M * v1 = M * a1 => v1 = a1 Velocity of second ball in case a: M * v2 = M * a2 => v2 = a2 Friction forces: a1 = -μk * g, a2 = μk * g b) Velocity of pallina in case b: m * vp = m * ap => vp = ap Velocity of second ball in case b: M * vr = M * ar => vr = ar Friction forces: ap = μk * g, ar = -μk * g Where: - a1, a2 = acceleration of balls in case a due to kinetic friction - ap, ar = acceleration of pallina and ball in case b due to kinetic friction - g = acceleration due to gravity (approximately 9.8 m/s²)

Calculate the final distances from the pallina

Now, we'll find the final distances of your two balls from the pallina using kinematic equations: a) Distance traveled by first ball: d1 = (1/2) * a1 * t1^2 Distance traveled by second ball: d2 = (1/2) * a2 * t2^2 b) Distance traveled by pallina: dp = (1/2) * ap * tp^2 Distance traveled by second ball: dr = (1/2) * ar * tr^2 Where t1, t2, tp, and tr are the time intervals balls and pallina take to come to rest. To find the time intervals, we can use the equation vf = vi + a*t. Since the final velocities of all objects will be zero, the time intervals can be calculated as t = -vi/a. Finally, to determine the final distances of the balls and pallina from their initial positions, we can add or subtract the distances traveled by them in each case: a) Final distance of first ball from pallina: 2.00 m + d1 Final distance of second ball from pallina: 2.00 m - d2 b) Final distance of pallina from the initial position: dp Final distance of second ball from pallina: dr + dp Remember to calculate the values for a1, a2, ap, ar, t1, t2, tp, tr, d1, d2, dp, and dr using the given values and equations before calculating the final distances for each case.

Key concepts

Conservation of Momentum

The conservation of momentum is a fundamental principle in physics stating that, within a closed system, the total momentum remains constant if no external forces are acting on it. Momentum, symbolized by the letter 'p', is the product of an object's mass (m) and its velocity (v). This concept is eloquently captured by the equation: \( p = m \times v \).

When applying this principle to collision problems, such as in the bocce ball scenario provided, we assert that the total momentum before the collision equals the total momentum after the collision. This means for:

• Case a) The momentum of the thrower's ball must be balanced by the sum of the momenta of both balls after the collision.
• Case b) The combined momentum of the thrower's ball and the stationary pallina must equal the post-collision momentum of both objects.

Understanding momentum conservation allows us to predict the behavior of objects post-collision, making it a powerful tool in physics problem-solving.

Kinetic Friction

Kinetic friction comes into play when two objects are in motion against each other. It is always opposite to the direction of movement and is quantified by the coefficient of kinetic friction (\( \mu_k \) ) which varies depending on the surfaces in contact. Frictional force (\( F_k \) ) is calculated using the equation: \( F_k = \mu_k \times N \), where N is the normal force — essentially, the force perpendicular to the surfaces in contact.

In our bocce problem, friction affects how far the balls roll after collision due to the force of kinetic friction acting against their motion. The frictional force opposes the movement, causes a deceleration (\(a = -\mu_k \times g\)), and eventually brings the balls to a stop. Calculating kinetic friction is vital to determine the final positions of the balls after the throw, factoring in how the surface resistance modifies their trajectories.

Kinematic Equations

Kinematic equations enable us to predict the future position and velocity of an object moving under constant acceleration. They connect the dots between an object's initial and final positions, velocities, and the acceleration affecting it over time. These equations include expressions such as \( d = v_i \times t + \frac{1}{2}a\times t^2 \) and \( v_f = v_i + a\times t \) where:\

• \(d\) is the displacement
• \(v_i\) is the initial velocity
• \(v_f\) is the final velocity
• \(a\) is the acceleration
• \(t\) is the time

For the bocce ball problem, these equations are crucial for determining how the friction-induced deceleration affects the distances travelled by the balls following the collision, which ultimately helps in calculating their final distances from the pallina. It's important to factor in that the final velocity (\(v_f\)) in this scenario will be zero, as the balls come to rest after rolling some distance due to friction.