Question

Very large accelerations can injure the body, especially if they last for a considerable length of time. The severity index (SI), a measure of the likelihood of injury, is defined as SI = a5/2t, where a is the acceleration in multiples of g and t is the time the acceleration lasts (in seconds). In one set of studies of rear end collisions, a person's velocity increases by 12 km/h with an acceleration of 35 m/s2.(a) What is the severity index for the collision? (s)(b) How far does the person travel during the collision if the car was initially moving forward at 7.0 km/h? (m)

Answer 1

a) The severity index (SI) is 3047.749, b) The injured travels 0.345 meters during the collision.

Step-by-step explanation:

a) The g-multiple of the acceleration, that is, a ratio of the person's acceleration to gravitational acceleration, is:

`a' = (35\,(m)/(s^(2)) )/(9.807\,(m)/(s^(2)) )`

`a' = 3.569`

The time taken for the injured to accelerate to final speed is given by this formula under the assumption of constant acceleration:

`v_(f) = v_(o) + a \cdot t`

Where:

`v_(o)` - Initial speed, measured in meters per second.

`v_(f)` - Final speed, measured in meter per second.

`a` - Acceleration, measured in
`(m)/(s^(2))`.

`t` - Time, measured in seconds.

`t = (v_(f)-v_(o))/(a)`

`t = (\left(12\,(km)/(h) \right)\cdot \left(1000\,(m)/(km) \right)\cdot \left((1)/(3600)\,(h)/(s) \right))/(35\,(m)/(s^(2)) )`

`t = 0.095\,s`

Lastly, the severity index is now determined:

`SI = (a'^(5))/(2\cdot t)`

`SI = (3.569^(5))/(2\cdot (0.095\,s))`

`SI = 3047.749`

b) The initial and final speed of the injured are
`1.944\,(m)/(s)` and
`5.278\,(m)/(s)`, respectively. The travelled distance can be determined from this equation of motion:

`v_(f)^(2) = v_(o)^(2) + 2\cdot a \cdot \Delta s`

Where
`\Delta s` is the travelled distance, measured in meters.

`\Delta s = (v_(f)^(2)-v_(o)^(2))/(2\cdot a)`

`\Delta s = (\left(5.278\,(m)/(s) \right)^(2)-\left(1.944\,(m)/(s) \right)^(2))/(2\cdot \left(35\,(m)/(s^(2)) \right))`

`\Delta s = 0.345\,m`.