A 1600 kg car traveling at a speed of 31 m/s skids to a halt on wet concrete where uk = 0.60. how long are the skid marks?

To find the length of the skid marks, we need to first calculate the deceleration of the car.

The deceleration can be found using the equation:
\(a = \frac{{v^2}}{{2d}}\)

Where:
\(a\) = acceleration (deceleration in this case)
\(v\) = initial velocity (31 m/s)
\(d\) = distance (skid marks)

Rearranging the equation, we get:
\(d = \frac{{v^2}}{{2a}}\)

The deceleration can be calculated using the equation:
\(a = g \cdot \mu_k\)

Where:
\(g\) = acceleration due to gravity (9.8 m/s^2)
\(\mu_k\) = coefficient of kinetic friction (0.60)

Substituting the values into the equation, we get:
\(a = 9.8 \cdot 0.60\)

Now, we can calculate the length of the skid marks using the equation:
\(d = \frac{{v^2}}{{2a}}\)

Substituting the values into the equation, we get:
\(d = \frac{{31^2}}{{2 \cdot (9.8 \cdot 0.60)}}\)

Now we can calculate the length of the skid marks.

To find the length of the skid marks, we can use the concept of work and energy.

The work done on the car will be equal to the change in kinetic energy. When the car skids to a halt, it loses all its kinetic energy.

The work done on the car is given by the equation:
Work = Force x Distance

In this case, the force opposing the motion of the car is the frictional force, which can be calculated using the coefficient of kinetic friction (uk) and the weight of the car.

The frictional force can be calculated using the equation:
Force of friction = coefficient of friction (uk) x Normal force

The normal force (Fn) is the force exerted by the surface on the car and is equal to the weight (mass x gravity) for an object on a horizontal surface.

Using Newton's second law of motion (F = ma), we can calculate the acceleration of the car:
Acceleration = Force of friction / mass

Once we have the acceleration, we can use the equation of motion (v² = u² + 2as), where u is the initial velocity, v is the final velocity (0 in this case), a is the acceleration, and s is the distance (skid marks) to solve for s.

Let's plug in the given values to calculate the skid marks:

Mass of the car (m) = 1600 kg
Initial velocity (u) = 31 m/s
Coefficient of kinetic friction (uk) = 0.60
Gravity (g) = 9.8 m/s²

Weight of the car (mg) = mass x gravity
Normal force (Fn) = Weight of the car

Force of friction = coefficient of friction x Normal force
Acceleration (a) = Force of friction / mass
Distance (skid marks) = (final velocity² - initial velocity²) / (2 x acceleration)

Weight of the car (mg) = 1600 kg x 9.8 m/s²
Normal force (Fn) = Weight of the car
Force of friction = 0.60 x Normal force
Acceleration (a) = Force of friction / mass
Distance (skid marks) = (0² -31²) / (2 x acceleration)

Now, let's do the calculations:

Weight of the car = 1600 kg x 9.8 m/s² = 15680 N
Normal force = Weight of the car = 15680 N
Force of friction = 0.60 x Normal force = 0.60 x 15680 N
Acceleration = Force of friction / mass = (0.60 x 15680 N) / 1600 kg
Distance (skid marks) = (0² -31²) / (2 x acceleration)

Now we can plug in the numbers and solve for the distance:

Acceleration = (0.60 x 15680 N) / 1600 kg
Distance (skid marks) = (0² -31²) / (2 x acceleration)

After performing the calculations, we find that the length of the skid marks is approximately 74.2 meters.

1/2 mv^2=mu*mg*distance