A 52-kg pole vaulter running at 12 m/s vaults over the bar. Her speed when she is above the bar is 0.7 m/s. Neglect air resistance, as well as any energy absorbed by the pole, and determine her altitude as she crosses the bar.
hmax = (V^2-Vo^2)/2g.
hmax = 0-144/-19.6 = 7.35 m.
V = Vo + gt.
t = (V-Vo)/g = (0.7-0)/9.8 = 0.0714 s.
h = ho - 0.5g*t^2'
h = 7.35 - 4.9*(0.0714)^2 = 7.325 m. =
her Alt.
If an automobile engine delivers 48.2 hp of power, how much time will it take for the engine to do 6.80 ✕ 105 J of work? (Hint: Note that one horsepower, 1 hp, is equal to 746 watts.)
To solve this problem, we can use the principle of conservation of mechanical energy. The mechanical energy of the vaulter will be conserved during the motion.
The mechanical energy (E) of an object is given by the sum of its kinetic energy (KE) and its potential energy (PE):
E = KE + PE
Initially, when the vaulter is running, all her energy is in the form of kinetic energy, as there is no height elevation yet. When she is above the bar, all her energy is in the form of potential energy, as her speed is reduced to zero. Therefore, we can write the equation as:
Initial kinetic energy (KE_initial) = Final potential energy (PE_final)
The initial kinetic energy is given by:
KE_initial = (1/2) * m * v^2
Where:
m = mass of the vaulter = 52 kg
v = velocity of the vaulter while running = 12 m/s
The final potential energy is given by the gravitational potential energy equation:
PE_final = m * g * h
Where:
g = acceleration due to gravity = 9.8 m/s^2
h = height of the vaulter above the bar (altitude)
Now we can equate the initial kinetic energy and final potential energy to solve for the altitude (h):
(1/2) * m * v^2 = m * g * h
Let's substitute the given values and solve for h:
(1/2) * 52 kg * (12 m/s)^2 = 52 kg * 9.8 m/s^2 * h
26 * 144 = 509.6 * h
3744 = 509.6 * h
h = 3744 / 509.6
h ≈ 7.34 meters.
Therefore, the vaulter's altitude as she crosses the bar is approximately 7.34 meters.
To determine the altitude of the pole vaulter as she crosses the bar, we can make use of the principle of conservation of mechanical energy. The mechanical energy of the vaulter consists of kinetic energy and potential energy.
The initial kinetic energy (KEi) of the vaulter is given by:
KEi = (1/2) * mass * (initial velocity)^2
The final kinetic energy (KEf) of the vaulter when she is above the bar is given by:
KEf = (1/2) * mass * (final velocity)^2
Since we are neglecting air resistance and the energy absorbed by the pole, the total mechanical energy remains constant throughout the motion. Therefore, the initial mechanical energy (MEi) is equal to the final mechanical energy (MEf).
The initial mechanical energy (MEi) is the sum of the initial kinetic energy (KEi) and the initial potential energy (PEi):
MEi = KEi + PEi
The final mechanical energy (MEf) is the sum of the final kinetic energy (KEf) and the final potential energy (PEf):
MEf = KEf + PEf
Since the pole vaulter is initially running, the initial potential energy (PEi) is negligible compared to the initial kinetic energy. Similarly, since the vaulter is above the bar, the final kinetic energy (KEf) becomes negligible compared to the final potential energy. Hence, we can simplify the equations as follows:
MEi = KEi
MEf = PEf
Now let's substitute the expressions for kinetic energy into the above equations:
MEi = (1/2) * mass * (initial velocity)^2
MEf = mass * g * altitude
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Since MEi = MEf, we can equate the two expressions:
(1/2) * mass * (initial velocity)^2 = mass * g * altitude
Cancelling the mass from both sides:
(1/2) * (initial velocity)^2 = g * altitude
Simplifying the equation:
altitude = [(1/2) * (initial velocity)^2] / g
Plugging in the values:
mass = 52 kg
initial velocity = 12 m/s
g = 9.8 m/s^2
altitude = [(1/2) * (12 m/s)^2] / 9.8 m/s^2
Calculating:
altitude ≈ 7.34 meters
Therefore, the altitude of the pole vaulter as she crosses the bar is approximately 7.34 meters.