During her high bar routine from question $2,$ Gabby Douglas slipped and falls from the high bar, landing on a $10$ cm thick gymnastics mat. She strikes the back of her head against the mat during her landing. Her head is moving at $8$ m$/$s when it first strikes the mat. The mass of her head is $6$ kg$.$ The impact ends when her head comes to a stop after deflecting the mat $6\mathrm{.}5$ cm$.$a$.$ How much kinetic energy does her head have the instant just before it hits the mat?b$.$ How much work does the mat do to stop the motion of her head?c$.$ What average impact force does the mat exert on her head during impact?d$.$ Estimate the peak impact force exerted by the mat on her head.e$.$ Estimate the peak acceleration of her head.f$.$ Express the peak acceleration in gs$.$