With what minimum speed must Fran leave the ground in order to lift her center of mass 1.90 m and cross the bar with a speed of 0.80 m/s?
her initial KE= change in GPE
+ final KE
1/2 m v^2=mgh+ 1/2 m (.8^2)
To determine the minimum speed Fran must leave the ground with in order to lift her center of mass to a height of 1.90 m and cross the bar with a speed of 0.80 m/s, we can make use of the principle of conservation of mechanical energy.
The principle of conservation of mechanical energy states that the total mechanical energy of a system remains constant as long as there are no external forces doing work on the system. In this case, we can assume that the only forces acting on Fran are gravity and the normal force from the ground, which are both conservative forces.
The total mechanical energy of Fran can be calculated as the sum of her kinetic energy and potential energy:
E = KE + PE
Since Fran is starting from the ground, her initial potential energy is zero. Therefore, the total mechanical energy at the start is just her initial kinetic energy:
E_initial = KE_initial
At the highest point of her jump, when her center of mass is at a height of 1.90 m, her kinetic energy is zero and her potential energy is at its maximum:
E_final = PE_final
Since no work is done by external forces, the total mechanical energy is conserved:
E_initial = E_final
Therefore, her initial kinetic energy is equal to her final potential energy:
KE_initial = PE_final
The initial kinetic energy can be expressed as:
KE_initial = (1/2)mv_initial²
Where m is the mass of Fran and v_initial is her initial velocity.
The final potential energy can be calculated as:
PE_final = mgh
Where g is the acceleration due to gravity (approximately 9.8 m/s²) and h is the height through which Fran's center of mass is lifted (1.90 m).
Substituting these values into the conservation equation, we get:
(1/2)mv_initial² = mgh
Simplifying the equation by canceling out the mass, we have:
(1/2)v_initial² = gh
Finally, to find the minimum speed Fran must leave the ground with, we rearrange the equation to solve for v_initial:
v_initial = √(2gh)
Substituting the given values, we can calculate the minimum speed:
v_initial = √(2 * 9.8 m/s² * 1.90 m)
v_initial ≈ 6.21 m/s
Therefore, Fran must leave the ground with a minimum speed of approximately 6.21 m/s in order to lift her center of mass 1.90 m and cross the bar with a speed of 0.80 m/s.