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
μK = 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
Well, you know what they say about friction - it's always trying to slow things down! Now, let's calculate the final speed of our block despite the friction trying to cramp our style.
First, let's determine the work done by the applied force on the block. The work done is given by the formula W = F * d * cos(theta), where W is the work done, F is the force applied, d is the distance, and theta is the angle between the force and the direction of motion. Since the force and motion are in the same direction, cos(theta) will be 1, making our life a little easier.
The work done is then W = 30 N * 18 m = 540 J.
Next, let's determine the work done by the friction force. The work done by friction is given by the negative product of the friction force and the distance, which means it's trying to suck away energy from our system.
The friction force is calculated by multiplying the coefficient of kinetic friction, which in this case is μK = 0.25, by the normal force. Since the block is on a level floor, the normal force is equal to the weight of the block, which is mass times gravity. In this case, the mass is 2.5 kg and the acceleration due to gravity is approximately 9.8 m/s^2.
So, the friction force is Ff = 0.25 * (2.5 kg * 9.8 m/s^2) = 6.125 N.
The work done by friction is then Wf = - Ff * d = -6.125 N * 18 m = -110.25 J.
Now, let's take into account the kinetic energy of our block. The work done on the block is equal to the change in kinetic energy. Therefore, the total work done is the sum of the work done by the applied force and the work done by friction.
W_total = W + Wf = 540 J + (-110.25 J) = 429.75 J.
The change in kinetic energy is then ΔKE = W_total = 429.75 J.
The change in kinetic energy is also equal to the final kinetic energy minus the initial kinetic energy. Since the initial speed is 2.0 m/s, the initial kinetic energy is KE_initial = (1/2) * 2.5 kg * (2.0 m/s)^2 = 10 J.
Therefore, the final kinetic energy is KE_final = KE_initial + ΔKE = 10 J + 429.75 J = 439.75 J.
Finally, we can determine the final speed of the block using the formula KE_final = (1/2) * mass * v^2, where v is the final speed we're looking for.
Solving for v, we have v = sqrt((2 * KE_final) / mass) = sqrt((2 * 439.75 J) / 2.5 kg) ≈ 9.08 m/s.
So, the final speed of the block, despite the cheeky friction, is approximately 9.08 m/s. Keep on rolling!
To find the final speed of the block, we need to consider the work done by the applied force and the work done by the friction force. Let's break down the steps:
Step 1: 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(theta), where theta is the angle between the force and the displacement.
In this case, the applied force is in the horizontal direction, and the displacement is also in the horizontal direction, so the angle between them is 0 degrees. Therefore, cos(theta) = 1.
The work done by the applied force is given by: work_applied = force_applied x distance.
Given: force_applied = 30 N, distance = 18 m.
Substituting the values, we get: work_applied = 30 N x 18 m = 540 Joules.
Step 2: Calculate the work done by the friction force.
The work done by the friction force is given by the formula: work_friction = force_friction x distance x cos(theta).
The friction force can be calculated using the formula: force_friction = μK x normal force.
The normal force is the force exerted by the horizontal surface on the block and is equal to the weight of the block, which is given by: normal force = mass x gravity.
Given: mass = 2.5 kg, μK = 0.25, gravity = 9.8 m/s^2.
Calculating the normal force: normal force = 2.5 kg x 9.8 m/s^2 = 24.5 N.
Calculating the force of friction: force_friction = 0.25 x 24.5 N = 6.125 N.
Now, substituting the values, we get: work_friction = force_friction x distance x cos(theta).
Given: distance = 18 m, cos(theta) = 1.
work_friction = 6.125 N x 18 m = 110.25 Joules.
Step 3: Calculate the net work done on the block.
The net work done on the block is the sum of the work done by the applied force and the work done by the friction force.
net work = work_applied - work_friction = 540 J - 110.25 J = 429.75 Joules.
Step 4: Use the work-energy theorem to find the final speed.
According to the work-energy theorem, the work done on an object equals the change in its kinetic energy.
The initial kinetic energy of the block is given by: initial KE = 1/2 x mass x initial velocity^2.
Given: mass = 2.5 kg, initial velocity = 2 m/s.
initial KE = 1/2 x 2.5 kg x (2 m/s)^2 = 5 Joules.
The final kinetic energy of the block is equal to the net work done on the block.
net work = final KE - initial KE.
Final KE = net work + initial KE = 429.75 J + 5 J = 434.75 Joules.
Finally, we can calculate the final speed using the equation: final KE = 1/2 x mass x final velocity^2.
Given: mass = 2.5 kg.
434.75 J = 1/2 x 2.5 kg x final velocity^2.
Simplifying, we get: final velocity^2 = 2 x (434.75 J) / (2.5 kg) = 347.8 m^2/s^2.
Taking the square root of both sides, we find: final velocity = sqrt(347.8 m^2/s^2) ≈ 18.64 m/s.
Therefore, the final speed of the block is approximately 18.64 m/s.
To find the final speed of the block, we need to calculate the work done on the block by the applied force and the work done by friction. The work done on an object is equal to the force applied multiplied by the distance traveled in the direction of the force.
First, let’s calculate the work done by the applied force. The formula for work is:
Work = Force × Distance
Given:
Applied force = 30 N
Distance = 18 m
Work done by the applied force = 30 N × 18 m = 540 J (joules)
Next, let’s calculate the work done by friction. The work done by friction is equal to the product of the force of friction and the distance traveled. The force of friction can be calculated using the formula:
Force of friction = μK × Normal force
The normal force is equal to the weight of the object, which can be calculated using the formula:
Weight = mass × gravitational acceleration
Given:
Mass = 2.5 kg
Gravitational acceleration = 9.8 m/s^2
Weight = 2.5 kg × 9.8 m/s^2 = 24.5 N
Now, we can calculate the force of friction:
Force of friction = 0.25 × 24.5 N = 6.125 N
Finally, we can calculate the work done by friction:
Work done by friction = Force of friction × Distance
Work done by friction = 6.125 N × 18 m = 110.25 J
Now that we have calculated the work done by the applied force and the work done by friction, we can find the change in the kinetic energy of the block.
Change in kinetic energy = Work by applied force - Work by friction
Change in kinetic energy = 540 J - 110.25 J = 429.75 J
The change in kinetic energy is also equal to the final kinetic energy minus the initial kinetic energy:
Change in kinetic energy = (1/2) × mass × (final velocity^2 - initial velocity^2)
Rearranging the equation to solve for the final velocity:
Final velocity = square root of [(Change in kinetic energy / (1/2) × mass)] + initial velocity^2
Final velocity = sqrt[(429.75 J / (1/2) × 2.5 kg)] + (2.0 m/s)^2
Final velocity = sqrt(343.8 m^2/s^2) + 4.0 m^2/s^2
Final velocity ≈ sqrt(347.8 m^2/s^2)
Final velocity ≈ 18.64 m/s
Therefore, the final speed of the block is approximately 18.64 m/s.