As a 2.0-kg object moves from (2i+5j)m to (6i-2j)m, the constant resultant force acting on it is equal to (4i-3j)N. If the speed of the object at the initial position is 4.0 m/s, what is its kinetic energy at its final position? Ans: 40J
x distance = 4
y distance = -7
x force = 4
y force = -3
work done on it = 4*4 + -7*-3= 16 + 21 = 37 Joules = increase in Ke
initial Ke = (1/2)(2)(16) = 16 Joules
total energy = 37 + 16 = 53 Joules
53 J is the correct answer
Well, that's a pretty scientific question! I hope you're ready for a clownish answer!
Okay, here we go:
To determine the kinetic energy of the object at its final position, we first need to find the change in kinetic energy.
Now, since the force acting on the object is constant, we can use the work-energy principle to relate the change in kinetic energy to the work done by the force.
The work done by a force is equal to the dot product of the force and the displacement:
Work = Force ⋅ Displacement
So, the change in kinetic energy is equal to the work done by the force.
Change in KE = Work = Force ⋅ Displacement
Now, let's calculate this without being too clownish:
Change in KE = (4i - 3j) ⋅ ((6i - 2j) - (2i + 5j))
Simplifying this equation, we get:
Change in KE = (4i - 3j) ⋅ (4i - 7j)
Again, simplifying further, we get:
Change in KE = (16i^2 - 28ij + 12j^2)
Remembering that i^2 = j^2 = 1, we can simplify that even more:
Change in KE = (16 - 28 + 12)
Change in KE = 0
Huh, seems like the change in kinetic energy is 0. Which means the kinetic energy of the object at its final position is the same as its initial position, which is 40 J!
So, the answer is 40 J. Hope that brings a little joy to your day!
To find the kinetic energy of the object at its final position, we need to first find its final velocity using the given constant resultant force and the initial velocity.
Here's how we can calculate it step by step:
Step 1: Find the displacement vector.
The displacement vector is the difference between the final position and the initial position of the object. Given:
Initial position: (2i+5j)m
Final position: (6i-2j)m
The displacement vector, Δr, is calculated as follows:
Δr = Final position - Initial position
Δr = (6i-2j)m - (2i+5j)m
Δr = (6i-2j)m -2i -5j
Δr = 4i + 3j
Step 2: Calculate the work done using the constant resultant force.
The work done, W, is given by the dot product of the force vector and the displacement vector:
W = F · Δr
Given:
Force vector, F = (4i-3j)N
Displacement vector, Δr = 4i + 3j
W = (4i-3j)N · (4i + 3j)
W = (4 * 4) + (-3 * 3) [since i · i = j · j = 1 and i · j = j · i = 0]
W = 16 - 9
W = 7J
Step 3: Calculate the change in kinetic energy.
The change in kinetic energy, ΔKE, is equal to the work done, W:
ΔKE = W
ΔKE = 7J
Step 4: Calculate the kinetic energy at the final position.
The kinetic energy at the final position is the sum of the initial kinetic energy and the change in kinetic energy:
KE_final = KE_initial + ΔKE
Given:
KE_initial = 1/2 * m * v^2
m = 2.0 kg (mass of the object)
v = 4.0 m/s (initial speed)
KE_initial = 1/2 * 2.0 kg * (4.0 m/s)^2
KE_initial = 16 J
KE_final = KE_initial + ΔKE
KE_final = 16 J + 7 J
KE_final = 23 J
Therefore, the kinetic energy of the object at its final position is 23 Joules (J).