A 1340 kg car going 21 m/s has to stop suddenly. The driver locks the brakes, and the car skids to a halt in a distance of 67 m .
What was the car's acceleration while stopping?
How much work was done by friction to stop the car?
What is the coefficient of kinetic friction between tires and road?
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To find the car's acceleration while stopping, we can use the formula for acceleration: acceleration (a) = (final velocity (v) - initial velocity (u)) / time (t). In this case, since the car came to a halt, its final velocity will be zero (v = 0). The initial velocity is given as 21 m/s (u = 21 m/s). Since we don't know the time taken to stop, we need to find it first.
To find the time taken to stop, we can use the formula: distance (d) = (initial velocity (u) × time (t)) + (0.5 × acceleration (a) × time (t)^2). In this case, the distance is given as 67 m (d = 67 m). Rearranging the formula, we get: 0.5 𝑎𝑡^2 + 𝑢𝑡 − 𝑑 = 0.
Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / 2a, where a = 0.5, b = u = 21 m/s, and c = -d = -67 m, we can solve for time. Taking only the positive root, we find t ≈ 4.35 seconds.
Now that we know the time taken to stop, we can calculate the acceleration: a = (v - u) / t = (0 - 21 m/s) / 4.35 s ≈ -4.83 m/s^2.
The negative sign indicates that the car is decelerating in the opposite direction to its initial motion.
Next, let's calculate the work done by friction to stop the car. The work done (W) is given by the formula: W = force (F) × distance (d). In this case, the force of friction is responsible for stopping the car.
The force of friction (F) can be calculated using the equation: F = mass (m) × acceleration (a). The mass of the car is given as 1340 kg (m = 1340 kg), and the acceleration calculated earlier is -4.83 m/s^2.
Substituting these values, we have: F = 1340 kg × (-4.83 m/s^2) ≈ -6466.2 N.
Since the work done (W) is equal to the force (F) multiplied by the distance (d), we can calculate: W = F × d = -6466.2 N × 67 m ≈ -432938.6 J.
The negative sign indicates that work is done against the motion of the car.
Finally, let's find the coefficient of kinetic friction (μ). The coefficient of kinetic friction can be calculated using the equation: μ = force of friction / normal force.
Since we don't know the normal force, we need to calculate it using the formula: normal force = mass (m) × acceleration due to gravity (g).
The acceleration due to gravity is approximately 9.8 m/s^2. Substituting the given values, we get: normal force = 1340 kg × 9.8 m/s^2 ≈ 13132 N.
Now, we can calculate the coefficient of kinetic friction: μ = force of friction / normal force = -6466.2 N / 13132 N ≈ -0.492.
The negative sign indicates that the frictional force acts opposite to the direction of motion of the car.
Therefore, the car's acceleration while stopping is approximately -4.83 m/s^2, the work done by friction to stop the car is approximately -432938.6 J, and the coefficient of kinetic friction between the tires and the road is approximately -0.492.