A car of mass 1300 kg is driving along a road at speed of 28m/s. As it approaches some traffic lights, the lights change to red and the driver applies his brakes causing the car to skid. The frictional force between the skidding tyres and the road is 0.8 of the weight of the car. Caculate how far the car will travel before it stops?
vf^2=vi^2 + 2ad
but a=-force/mass=mg*mu/m =0 mu*g
so
vf^2=0=vi^2- 2mu/g*distance
solve for distance, mu is given, vi is given
To calculate the distance the car will travel before it stops, we can use the concept of work and energy.
Step 1: Convert the mass of the car from kg to Newtons (N).
The weight of the car can be calculated using the formula:
Weight = mass * gravity
where the mass is 1300 kg and gravity is approximately 9.8 m/s^2.
Weight = 1300 kg * 9.8 m/s^2 = 12,740 N
Step 2: Calculate the frictional force between the skidding tires and the road.
The frictional force is 0.8 times the weight of the car, so we can calculate it as follows:
Frictional Force = 0.8 * Weight = 0.8 * 12,740 N = 10,192 N
Step 3: Calculate the work done by the frictional force.
The work done is equal to the force applied multiplied by the distance traveled.
In this case, the force applied is the frictional force, and since the car is skidding, the work done is equal to the change in kinetic energy of the car.
Work Done = Change in Kinetic Energy
Change in Kinetic Energy = (1/2) * mass * final velocity^2 - (1/2) * mass * initial velocity^2
The final velocity is 0 m/s since the car comes to a stop, and the initial velocity is 28 m/s.
Change in Kinetic Energy = (1/2) * 1300 kg * (0 m/s)^2 - (1/2) * 1300 kg * (28 m/s)^2 = -254,800 J (negative because energy is lost)
Step 4: Calculate the distance traveled using work and the frictional force.
Work Done = Frictional Force * Distance
Distance = Work Done / Frictional Force
Distance = -254,800 J / 10,192 N = -24.95 m
The negative sign indicates that the direction is opposite to the motion of the car.
Therefore, the car will travel approximately 24.95 meters before it comes to a stop.
To calculate the distance the car will travel before it stops, we can use the equations of motion. The main equation we will use is the one that relates distance, initial velocity, final velocity, acceleration, and time:
\[v_f^2 = v_i^2 + 2a \cdot d\]
where \(v_f\) is the final velocity (zero in this case, as the car stops), \(v_i\) is the initial velocity (given as 28 m/s), \(a\) is the acceleration, and \(d\) is the distance.
First, let's calculate the deceleration of the car. The deceleration \(a\) is given by:
\[a = \frac{{\text{{Frictional force}}}}{{\text{{Mass}}}}\]
We are given that the frictional force is 0.8 times the weight of the car. The weight of the car is calculated as the mass multiplied by the acceleration due to gravity (which we can assume as 9.8 m/s^2):
\[Frictional\ force = 0.8 \times (9.8 \times \text{{Mass}})\]
Now, we can substitute the values we know into the equation to find the distance \(d\):
\[0 = (28^2) + 2 \cdot a \cdot d\]
Now we have two equations with two unknowns: \(a\) and \(d\). We can substitute the value of the frictional force we calculated earlier into the equation for \(a\):
\[0 = (28^2) + 2 \cdot \left(\frac{{0.8 \times (9.8 \times \text{{Mass}})}}{{\text{{Mass}}}}\right) \cdot d\]
Simplifying the equation gives:
\[0 = 784 + 15.68 \cdot d\]
Now solve for \(d\):
\[d = \frac{{-784}}{{15.68}}\]
Calculating the value gives:
\[d \approx -50\]
The negative sign indicates that the car stops after traveling in the negative direction. Since distance cannot be negative, we take the absolute value of \(d\):
\[d \approx 50\]
Therefore, the car will travel approximately 50 meters before it stops.