CP BIO Whiplash Injuries.? When a car is hit from behind, its passengers undergo sudden forward acceleration, which can cause a severe neck injury known as ?whiplash?. During normal acceleration, the neck muscles play a large role in accelerating the head so that the bones are not injured. But during a very sudden acceleration, the muscles do not react immediately because they are flexible; most of the accelerating force is provided by the neck bones. Experiments have shown that these bones will fracture if they absorb more than 8.0 J of energy. (a) If a car waiting at a stoplight is rear-ended in a collision that lasts for 10.0 ms, what is the greatest speed this car and its driver can reach without breaking neck bones if the driver’s head has a mass of 5.0 kg (which is about right for a 70-kg person)? Express your answer in m/s and in mi/h. (b) What is the acceleration of the passengers during the collision in part (a), and how large a force is acting to accelerate their heads? Express the acceleration in m/s2 and in g’s.

Solution 69P Step 1: a) The initial velocity of the person, u = 0 m/s The final velocity of the person, v = Mass of the head, m = 5 kg Maximum energy with which the bones can withstand, E = 8.0 J Therefore, we can write, the change in Kinetic energy during the collision, KE = 8.0 J That is, ½ mv - ½ mu = 8.0 J Or, ½ mv - 0 = 8.0 J v = 2 × 8.0 J / m = 16 J / 5 kg = 3.2 m /s 2 2 Taking square roots on both sides, v = 1.79 m/s 1 mile = 1609.34 m Therefore, 1 m = 1/1609.34 miles 1 hour = 3600 s 1 s = 1/3600 h Changing the units, v = 1.79 {(1/1609.34 )/(1/3600)} mi/hr v = 1.79 (3600/1609.34) mi/hr = 1.79 × 2.24 mi/hr = 4 mi/hr Step 2: b) rovided, the time for the collision, t = 10 ms 1 ms = 10 s-3 10 ms = 10×1 0 s = 10 s2 Take the initial velocity of the car, u = 1.79 m/s Final velocity of the car, v = 0 m/s (it comes to rest after the collision) v = u+at (Newton’s equations of motion) Substituting the values in the equation, -2 0 = 1.79 m/s + (a × 10 s) -2 - 1.79 m/s = (a × 10 s) Rearranging the equation, a = - 1.79 m/s / 10 s -2 a = - 179 m/s . this is the acceleration (deceleration) experienced by the passenger.