You throw a 20 N rock into the air from ground level and observe that when it is 1 5.0 M high, it is traveling upward at 25.0 m/s. Use the work energy principle to find (a) the rock’s speed just as it left the ground and (b) the maximum height the rock will reach.
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To find the rock's speed just as it left the ground and the maximum height it will reach, we can use the work-energy principle.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy, which can be expressed as:
Work = ΔKE
First, let's calculate the initial kinetic energy of the rock just as it left the ground.
Given:
Force (F) = 20 N
Height (h) = 5.0 m
Velocity (v_1) = 25.0 m/s
We can calculate the work done on the rock using the formula:
Work = Force × Distance
Work = 20 N × 5.0 m
Work = 100 J
According to the work-energy principle, this work done is equal to the change in kinetic energy (ΔKE). The initial kinetic energy (KE_1) can be expressed as:
KE_1 = Work
KE_1 = 100 J
Now, let's find the final kinetic energy of the rock when it reached the highest point (maximum height).
At the highest point, the rock's velocity (v_2) will be zero since it momentarily stops moving before it starts to fall back down. Thus, we have:
Final kinetic energy (KE_2) = 0 J
Now, we can find the maximum height reached by equating the initial kinetic energy (KE_1) to the potential energy (PE) at the highest point.
PE = m × g × h
Given:
Force (F) = 20 N
Height (h) = 5.0 m
Using the definition of work, PE at the highest point can be calculated as:
PE = Work = Force × Distance
PE = 20 N × 5.0 m
PE = 100 J
Since potential energy at the highest point is equal to the initial kinetic energy, we have:
PE = KE_1
100 J = KE_1
Now, we have the value for initial kinetic energy (KE_1) and the value for final kinetic energy (KE_2).
Lastly, we can use the fact that the change in kinetic energy (ΔKE) is equal to the difference between the final and initial kinetic energies:
ΔKE = KE_2 - KE_1
Since KE_2 = 0 J and KE_1 = 100 J, we get:
ΔKE = -100 J
The negative sign indicates that the change in kinetic energy is a decrease from the initial kinetic energy.
(a) The rock's speed just as it left the ground is calculated using the initial kinetic energy (KE_1):
KE_1 = (1/2) × m × (velocity_1)^2
100 J = (1/2) × m × (25.0 m/s)^2
Solving for mass (m):
100 J = (1/2) × m × 625.0 m^2/s^2
m = 100 J / (1/2 × 625.0 m^2/s^2)
m = 0.32 kg
Using the obtained mass, we can calculate the speed (v_0) just as the rock left the ground:
KE_1 = (1/2) × m × (v_0)^2
100 J = (1/2) × 0.32 kg × (v_0)^2
Solving for v_0:
200 J = 0.32 kg × (v_0)^2
v_0^2 = 200 J / 0.32 kg
v_0^2 = 625 m^2/s^2
v_0 = √625 m^2/s^2
v_0 = 25 m/s
Therefore, the rock's speed just as it left the ground is 25 m/s.
(b) The maximum height reached by the rock can be found using the potential energy equation:
PE = m × g × h
Given:
Force (F) = 20 N
Height (h) = unknown
Using the definition of work, PE at the highest point can be calculated as:
PE = 20 N × h
Since the potential energy at the highest point is equal to the initial kinetic energy (KE_1):
20 N × h = 100 J
Solving for height (h):
h = 100 J / (20 N)
h = 5.0 m
Therefore, the maximum height reached by the rock is 5.0 m.