In the high jump, Fran's kinetic energy is transformed into gravitational potential energy without the aid of a pole.
With what minimum speed must Fran leave the ground in order to lift her center of mass 1.95 m and cross the bar with a speed of 0.70 m/s?
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Her initial kinetic energy must be greater than her kinetic energy at the top, by an amount equal to m g H, where H = 1.95 m.
The m's cancel out when you write the kinetic energy terms.
To determine the minimum speed Fran must leave the ground with in order to achieve a desired height and cross the bar with a desired speed, we can use 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 no external forces are acting on it. In this case, we can assume there are no significant external forces acting on Fran during the high jump.
The total mechanical energy of an object is the sum of its kinetic energy (KE) and potential energy (PE). In this case, Fran's kinetic energy is transformed into gravitational potential energy.
Let's break down the problem step by step:
Step 1: Determine Fran's kinetic energy at takeoff
The initial kinetic energy will be transformed into gravitational potential energy. We can use the equation for kinetic energy:
KE = 0.5 * m * v^2
Where:
KE is the kinetic energy
m is the mass of Fran
v is the velocity of Fran at takeoff
Since the mass of Fran is not provided, we can assume it cancels out when comparing the initial and final kinetic energies. Therefore, we can proceed without explicitly knowing Fran's mass.
Step 2: Determine the gravitational potential energy at the peak of the jump.
At the peak of her jump (when her center of mass has reached a height of 1.95 m), Fran will have achieved maximum height and zero velocity. At this point, all of her initial kinetic energy will have been converted into gravitational potential energy. The equation for gravitational potential energy is:
PE = m * g * h
Where:
PE is the gravitational potential energy
m is the mass of Fran
g is the acceleration due to gravity (9.8 m/s^2)
h is the height reached by Fran's center of mass (1.95 m in this case)
Step 3: Calculate the final kinetic energy as Fran crosses the bar.
The final kinetic energy of Fran as she crosses the bar is given as 0.70 m/s.
Step 4: Apply the principle of conservation of mechanical energy.
According to the principle of conservation of mechanical energy, the total mechanical energy at takeoff should be equal to the total mechanical energy at the peak of the jump.
Initial kinetic energy (KE) = Gravitational potential energy (PE) + Final kinetic energy (KE)
Equating the expressions for kinetic energy and gravitational potential energy, we can solve for the minimum velocity (v) at takeoff.
0.5 * m * v^2 = m * g * h + 0.5 * m * (0.70 m/s)^2
Simplifying and canceling out the mass (m):
0.5 * v^2 = g * h + 0.5 * (0.70 m/s)^2
Now we can solve for v:
v^2 = 2 * (g * h + 0.5 * (0.70 m/s)^2)
v = sqrt(2 * (g * h + 0.5 * (0.70 m/s)^2))
Plugging in the known values:
v = sqrt(2 * (9.8 m/s^2 * 1.95 m + 0.5 * (0.70 m/s)^2))
Evaluating the equation will give us the minimum speed Fran must leave the ground with in order to achieve the desired result.