A box of unknown mass is sliding with an initial speed vi = 5.40 m/s across a horizontal friction-less warehouse floor when it encounters a rough section of flooring d = 3.50 m long. The coefficient of kinetic friction between the rough section of flooring and the box is 0.100. Using energy considerations, determine the final speed of the box after sliding across the rough section of flooring. _____________ m/s
initial Ke = (1/2) m * 5.4^2
energy lost to friction
= 0.1 * m g * 3.50
final ke = initial ke - lost energy
(1/2) m v^2 = (1/2) m * 5.4^2 - 0.1 * m g * 3.50
solve for v
note that the mass cancels
Well, well, well, isn't this a slippery situation? Let's slide into the solution, shall we?
We can use the principle of conservation of energy to tackle this problem. The initial kinetic energy (KE) of the box will convert into work done against friction, which will consume the energy, leaving us with the final kinetic energy.
The initial KE is given by the formula KE = (1/2)mv^2, where m is the mass of the box and v is the initial velocity. Since the mass is unknown, we'll just call it 'm' for now.
The work done against friction is given by W = μmgd, where μ is the coefficient of kinetic friction, m is the mass of the box (again), g is the acceleration due to gravity, and d is the distance.
Setting the initial KE equal to the work done against friction, we have (1/2)mv^2 = μmgd.
Now, let's solve for the final velocity. Rearranging the equation, we get v^2 = 2μgd. Taking the square root of both sides gives us v = √(2μgd).
Plugging in the values, we have v = √(2 * 0.100 * 9.8 * 3.50).
Evaluating that, we get v ≈ 3.15 m/s (rounded to two decimal points).
So, after all that sliding and friction, the final speed of the box is approximately 3.15 m/s. Keep on rollin'!
To determine the final speed of the box after sliding across the rough section of flooring, we can use the principle of conservation of energy.
1. The initial kinetic energy of the box is given by:
KE_initial = 0.5 * m * v_initial^2
where m is the mass of the box and v_initial is the initial speed of the box.
2. As the box slides across the rough section of flooring, some of its kinetic energy is converted into thermal energy due to friction. The work done by friction can be calculated as:
Work_friction = -μ * m * g * d
where μ is the coefficient of kinetic friction, m is the mass of the box, g is the acceleration due to gravity, and d is the distance over which the box slides across the rough section of flooring.
3. The final kinetic energy of the box can be calculated as:
KE_final = KE_initial - Work_friction
4. Finally, we can solve for the final speed of the box by rearranging the kinetic energy equation:
v_final = sqrt(2 * KE_final / m)
To find the final speed, we need the mass of the box. If you provide the mass, we can calculate the final speed.
To determine the final speed of the box after sliding across the rough section of flooring using energy considerations, we can consider the conservation of mechanical energy.
The initial kinetic energy of the box is given by:
KE_initial = (1/2) * m * v_initial^2
Where m is the mass of the box and v_initial is the initial speed of the box, which is 5.40 m/s.
The final kinetic energy can be expressed as:
KE_final = (1/2) * m * v_final^2
Where v_final is the final speed of the box, which is what we need to find.
The work done by friction is given by:
Work_friction = -μ * N * d
Where μ is the coefficient of kinetic friction, N is the normal force (equal to the weight of the box), and d is the distance over which the friction acts.
The work done by friction results in a decrease in the kinetic energy of the box. Therefore, we can write:
Work_friction = KE_initial - KE_final
Substituting the values, we have:
-μ * N * d = (1/2) * m * v_initial^2 - (1/2) * m * v_final^2
The normal force N can be calculated using the weight of the box:
N = m * g
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting this into the previous equation, we now have:
-μ * m * g * d = (1/2) * m * v_initial^2 - (1/2) * m * v_final^2
Now we can rearrange the equation to solve for v_final:
(1/2) * m * v_final^2 = (1/2) * m * v_initial^2 + μ * m * g * d
Canceling out the mass term and rearranging further, we get:
v_final^2 = v_initial^2 + 2 * μ * g * d
Finally, taking the square root of both sides, we find:
v_final = sqrt(v_initial^2 + 2 * μ * g * d)
Now we can substitute the values given in the problem:
v_final = sqrt(5.40^2 + 2 * 0.100 * 9.8 * 3.50)
Calculating this expression, we find:
v_final ≈ 4.98 m/s
Therefore, the final speed of the box after sliding across the rough section of flooring is approximately 4.98 m/s.