A horizontal force of 30 N pulls a block of mass 2.5 kg across a level floor. The coefficient of kinetic friction between the block and the floor is
ìK = 0.25.
If the block begins with a speed of 2.0 m/s and is pulled for a distance of 18 m, what is the final speed of the block? ____m/s
To find the final speed of the block, we need to consider the work done by the applied force and the work done by friction.
First, let's calculate the work done by the applied force. The work done by a force is given by the formula:
Work = Force x Distance x cos(θ)
In this case, the force is horizontal (parallel to the displacement), so θ (the angle between the force and displacement) is 0 degrees. The work done by the applied force is:
Work_applied = Force x Distance = 30 N x 18 m = 540 J
Next, let's calculate the work done by friction. The work done by friction is given by the formula:
Work_friction = Force_friction x Distance x cos(180°)
The force of friction can be calculated using the formula:
Force_friction = μK * Normal Force
The normal force is equal to the weight of the block, which can be calculated as:
Weight = mass x gravity
Given that the mass of the block is 2.5 kg and the acceleration due to gravity is approximately 9.8 m/s^2, the weight of the block is:
Weight = 2.5 kg x 9.8 m/s^2 = 24.5 N
Using the coefficient of kinetic friction μK = 0.25, we can now calculate the force of friction:
Force_friction = 0.25 x 24.5 N = 6.125 N
Since the force of friction acts opposite to the direction of motion, the angle between the force of friction and displacement is 180 degrees.
Now, we can calculate the work done by friction:
Work_friction = Force_friction x Distance x cos(180°) = 6.125 N x 18 m x cos(180°) = -110.25 J (note the negative sign indicating work being done against the motion)
Finally, to find the final speed of the block, we can use the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy.
The initial kinetic energy of the block is given by:
Initial Kinetic Energy = (1/2) x mass x (initial speed)^2
The final kinetic energy of the block is given by:
Final Kinetic Energy = (1/2) x mass x (final speed)^2
Since the work done by both the applied force and friction are equal to the change in kinetic energy, we can write the equation as follows:
Work_applied + Work_friction = Final Kinetic Energy - Initial Kinetic Energy
Plugging in the values we calculated:
540 J + (-110.25 J) = (1/2) x 2.5 kg x (final speed)^2 - (1/2) x 2.5 kg x (2.0 m/s)^2
Simplifying the equation:
429.75 J = (1/2) x 2.5 kg x (final speed)^2 - 5 J
Rearranging the equation to solve for the final speed:
(final speed)^2 = (429.75 J + 5 J) x (2/2.5 kg) = 175 J/kg
Taking the square root to find the final speed:
final speed = √(175 J/kg) ≈ 13.23 m/s
Therefore, the final speed of the block is approximately 13.23 m/s.