Final answer:
To find out if Gabrielle will clear the lake and make it to the forest in time, we calculate the potential energy in the bungee, the force of friction on the grass patch, and then determine if the kinetic energy is sufficient to travel 21 meters over the ice in 5.1 seconds.
Step-by-step explanation:
To determine if Gabrielle will clear the lake and make it into the forest within the 5.1 seconds, we need to calculate the forces at play and the resulting motion. First, we will calculate the potential energy stored in the bungee sling as Gabrielle stretches it. The potential energy (PE) stored in a stretched spring is given by PE = 1/2 * k * x^2, where k is the spring constant and x is the stretch distance. Replacing with the values given, we have PE = 1/2 * 97 N/m * (2.9 m)^2. Secondly, this potential energy is converted to kinetic energy (KE), which will be reduced due to the friction against the grass before reaching the lake. The force of friction (f) can be calculated using f = μ * m * g, where μ is the coefficient of friction, m is the combined mass of Gabrielle and the cart, and g is the acceleration due to gravity. This kinetic energy minus the work done aacgainst friction (f * distance over grass) will be the energy she has as she enters the frozen lake. Lastly, if we assume no friction on the ice, we can calculate whether the remaining kinetic energy is enough for her to cross the 21-meters-wide lake within 5.1 seconds.