A 0.422-kg ball with a speed of 1.31 m/s rolls across a level surface toward an open 0.348-kg box that is resting on its side. The ball enters the box, and the box (with the ball inside it) then slides across the surface a distance of 0.473 m. What is the coefficient of kinetic friction between the box and the table?
0.435
0.0629
To find the coefficient of kinetic friction between the box and the table, we can use the concept of conservation of momentum.
First, let's calculate the initial momentum of the ball before it enters the box. Momentum (p) is calculated by multiplying mass (m) and velocity (v):
Initial momentum of the ball = mass of the ball * velocity of the ball
= 0.422 kg * 1.31 m/s
= 0.55382 kg·m/s
Next, let's calculate the final momentum of the ball and the box together after they slide a distance of 0.473 m. Since momentum is conserved, the final momentum will be equal to the initial momentum. The final momentum is given by:
Final momentum of the ball and the box = (mass of the ball + mass of the box) * final velocity
By using the principle of conservation of momentum, the final momentum is equal to the momentum before the collision.
0.55382 kg·m/s = (0.422 kg + 0.348 kg) * final velocity
Simplifying the equation:
0.55382 kg·m/s = 0.77 kg * final velocity
Now we can calculate the final velocity:
final velocity = (0.55382 kg·m/s) / (0.77 kg)
≈ 0.719 m/s
Now, we need to calculate the acceleration of the box.
Using the formula for acceleration:
final velocity = initial velocity + acceleration * distance
0.719 m/s = 0 + acceleration * 0.473 m
Simplifying the equation:
acceleration = 0.719 m/s / 0.473 m
≈ 1.52 m/s²
The net force acting on the box is the force of kinetic friction, which can be calculated using Newton's second law:
Force of kinetic friction = mass of the box * acceleration
Force of kinetic friction = 0.348 kg * 1.52 m/s²
≈ 0.53296 N
Finally, we can find the coefficient of kinetic friction using the equation:
Force of kinetic friction = coefficient of kinetic friction * normal force
In this case, the normal force is the weight of the box, which is equal to the mass of the box multiplied by the acceleration due to gravity (9.8 m/s²):
0.53296 N = coefficient of kinetic friction * (0.348 kg * 9.8 m/s²)
Simplifying the equation:
coefficient of kinetic friction = 0.53296 N / (0.348 kg * 9.8 m/s²)
≈ 0.168
Therefore, the coefficient of kinetic friction between the box and the table is approximately 0.168.