A 67.0 kg base runner begins his slide into second base when moving at a speed of 3.5 m/s. The coefficient of friction between his clothes and Earth is 0.70. He slides so that his speed is zero just as he reaches the base.
How much mechanical energy is lost due to friction acting on the runner?
How far does he slide?
To determine the amount of mechanical energy lost due to friction acting on the runner, we need to find the work done by the friction force.
1. Calculate the work done by friction:
Work = Force x Distance
The friction force can be found using the equation:
Force of friction = coefficient of friction x normal force
The normal force can be calculated as:
Normal force = mass x gravity
Normal force = 67.0 kg x 9.8 m/s^2
Once we have the normal force, we can calculate the friction force:
Force of friction = 0.70 x (67.0 kg x 9.8 m/s^2)
2. Now, we need to determine the distance over which the work is done. The total distance the runner slides can be found using the equation of motion:
Velocity^2 = Initial velocity^2 + 2 x acceleration x distance
Since the final velocity is zero:
0^2 = (3.5 m/s)^2 + 2 x acceleration x distance
Rearranging the equation, we get:
Distance = (0 - (3.5 m/s)^2) / (2 x acceleration)
Acceleration can be calculated using:
Acceleration = Force of friction / mass
3. Substitute the calculated values into the equations to find the answer.
First, calculate the force of friction:
Force of friction = 0.70 x (67.0 kg x 9.8 m/s^2)
Then, calculate the distance:
Distance = (0 - (3.5 m/s)^2) / (2 x (Force of friction / mass))
Plug in the values and calculate the distance.
To determine how much mechanical energy is lost due to friction acting on the runner, you need to calculate the work done by friction.
The work done by friction can be calculated using the equation: work = force x distance x cos(theta), where force is the frictional force, distance is the distance over which the force acts, and theta is the angle between the force and the direction of motion.
First, let's find the frictional force. The frictional force is equal to the product of the coefficient of friction and the normal force. The normal force is equal to the weight of the runner, which can be calculated using the equation: weight = mass x gravity, where mass is the mass of the runner (67.0 kg) and gravity is the acceleration due to gravity (9.8 m/s^2).
So the normal force is: normal force = 67.0 kg x 9.8 m/s^2 = 656.6 N.
Therefore, the frictional force is: frictional force = coefficient of friction x normal force = 0.70 x 656.6 N = 459.6 N.
Next, we need to find the distance over which the frictional force acts. In this case, the distance is the distance over which the runner slides. To find this distance, we can use the equation of motion: vf^2 = vi^2 + 2ad, where vf is the final velocity (0 m/s), vi is the initial velocity (3.5 m/s), a is the acceleration (which is equal to the frictional force divided by the mass), and d is the distance.
Plugging in the known values, we have: (0 m/s)^2 = (3.5 m/s)^2 + 2(459.6 N/67.0 kg)d.
Simplifying, we get: 0 = 12.25 m^2/s^2 + 13.6791 m^2/s^2d.
Rearranging the equation, we have: d = -12.25 m^2/s^2 / (13.6791 m^2/s^2) = -0.8955 m^2.
Since distance cannot be negative, we discard the negative sign and the distance over which the runner slides is approximately 0.8955 meters.
Finally, to find the mechanical energy lost due to friction, we can use the work-energy principle. The work done by friction is equal to the change in mechanical energy, which can be calculated using the equation: work = change in kinetic energy.
The change in kinetic energy is equal to the initial kinetic energy minus the final kinetic energy. The initial kinetic energy can be calculated using the equation: kinetic energy = 0.5 x mass x velocity^2.
So the initial kinetic energy is: 0.5 x 67.0 kg x (3.5 m/s)^2 = 404.75 J.
The final kinetic energy is: 0.5 x 67.0 kg x (0 m/s)^2 = 0 J.
Therefore, the change in kinetic energy is: change in kinetic energy = 404.75 J - 0 J = 404.75 J.
Hence, the amount of mechanical energy lost due to friction acting on the runner is approximately 404.75 Joules.
(a) All of his (or her) initial kinetic energy is lost.
(b) Initial KE = (1/2) M V^2 = work done against friction = (M*g*Uk*X)
Uk = 0.7 is the coefficient of friction
Solve for X
X = V^2/(2*g*Uk)