In Sample Problems 4.2 item 2, find the speed of the object at,
A) x = 2.0 m
B) x = 3.0 m
C) x = 4.0 m
Assume that the mass of the object is 2.5 kg and that the object starts from rest.
Given:
m = 2.5 kg
v↓0 = 0
W = 25 J
A) x = 2.0 m:
v = √(2W/m) = √(2*25/2.5) = 5 m/s
B) x = 3.0 m:
v = √(2W/m) = √(2*25/2.5) = 5 m/s
C) x = 4.0 m:
v = √(2W/m) = √(2*25/2.5) = 5 m/s
To find the speed of an object at different positions, 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.
The work done on an object can be calculated using the formula:
W = ∫F * dx
Where W is the work done, F is the force acting on the object, and dx is the displacement of the object.
In this case, the work done is given as W = 25 J. Since the object starts from rest (v₀ = 0), the initial kinetic energy (KE₀) is zero.
Using the work-energy principle, we can equate the work done to the change in kinetic energy:
W = KE - KE₀
Since KE₀ = 0, the equation simplifies to:
W = KE
Now we can find the kinetic energy (KE) at different positions and then calculate the speed (v) using the formula:
KE = 0.5 * m * v²
Let's solve for each position:
A) x = 2.0 m:
For this position, we need to calculate the work done over the displacement from 0 to 2.0 m. Since the force acting is not given, we cannot directly calculate the work done. We need more information or a force-displacement relationship to proceed.
B) x = 3.0 m:
Similarly, we need additional information or a force-displacement relationship to calculate the work done at x = 3.0 m.
C) x = 4.0 m:
Again, we need additional information or a force-displacement relationship to calculate the work done at x = 4.0 m.
Without the necessary information, it is not possible to determine the speed of the object at each specific position.
To find the speed of the object at different positions, we need to use the work-energy principle and the equation for kinetic energy.
The work done on an object is equal to the change in its kinetic energy. Mathematically, this is expressed as:
W = ΔKE
where W represents the work done on the object and ΔKE represents the change in kinetic energy.
The work-energy principle can be rearranged to solve for the change in kinetic energy:
ΔKE = W
Since the object starts from rest, its initial kinetic energy is zero. Thus, the equation can be further simplified as:
KE = W
The equation for kinetic energy is:
KE = ½mv²
where KE represents the kinetic energy, m represents the mass of the object, and v represents its velocity.
We can equate the equations for kinetic energy and work:
½mv² = W
Now, we can solve for the velocity (v).
Step 1: Calculate the velocity (v) using the given values.
½mv² = W
Substitute the given values:
m = 2.5 kg
W = 25 J
½(2.5 kg)v² = 25 J
Step 2: Solve for v.
Divide both sides of the equation by ½(2.5 kg):
v² = (25 J) / (½(2.5 kg))
v² = 10 J/kg
Taking the square root of both sides:
v = √(10 J/kg)
Now we can find the speed of the object at different positions.
A) x = 2.0 m
B) x = 3.0 m
C) x = 4.0 m
Since the object starts from rest, its initial speed (v₀) is zero.
Using the equation of motion:
v² = v₀² + 2ax
where v represents the speed of the object, v₀ represents its initial speed, a represents its acceleration, and x represents the displacement.
Since the object starts from rest, the initial speed (v₀) is zero.
Let's calculate the speed of the object at different positions using the given values.
A) x = 2.0 m
v² = v₀² + 2ax
v² = 0 + 2(10 J/kg)(2.0 m)
v² = 0 + 40 J/kg·m
v = √(40 J/kg·m)
B) x = 3.0 m
v² = v₀² + 2ax
v² = 0 + 2(10 J/kg)(3.0 m)
v² = 0 + 60 J/kg·m
v = √(60 J/kg·m)
C) x = 4.0 m
v² = v₀² + 2ax
v² = 0 + 2(10 J/kg)(4.0 m)
v² = 0 + 80 J/kg·m
v = √(80 J/kg·m)
Now, you can calculate the numerical values for the speed of the object at x = 2.0 m, x = 3.0 m, and x = 4.0 m using the appropriate formulas.