a 2.00 kg object has a velocity of 4.00i -3.00j m/s.
What is the kinetic energy at this moment?
What is the net work done on the object if its velocity change 6.00i +2.00j m/s
25
To find the kinetic energy of an object, we can use the equation:
Kinetic Energy = 0.5 * mass * velocity^2
Given:
Mass (m) = 2.00 kg
Velocity (v) = 4.00i - 3.00j m/s
To calculate the kinetic energy, we need to find the magnitude of the velocity vector:
Velocity = 4.00i - 3.00j m/s
The magnitude of the velocity vector can be calculated using the Pythagorean theorem:
Magnitude of Velocity = √(v_{x}^2 + v_{y}^2)
where v_{x} and v_{y} represent the x and y components of the velocity vector, respectively.
For the given velocity:
v_{x} = 4.00 m/s
v_{y} = -3.00 m/s
So, the magnitude of the velocity vector is:
Magnitude of Velocity = √((4.00)^2 + (-3.00)^2) = √(16 + 9) = √25 = 5.00 m/s
Now, we can calculate the kinetic energy:
Kinetic Energy = 0.5 * mass * velocity^2
= 0.5 * 2.00 kg * (5.00 m/s)^2
= 0.5 * 2.00 kg * 25.00 m^2/s^2
= 25.00 Joules
Therefore, the kinetic energy of the object at this moment is 25.00 Joules.
Now let's move on to the second part of the question:
Given the change in velocity:
Δv = 6.00i + 2.00j m/s
To find the net work done on the object, we can use the work-energy theorem:
Net Work = Change in Kinetic Energy
The change in kinetic energy is given by:
Change in Kinetic Energy = 0.5 * mass * (final velocity^2 - initial velocity^2)
First, we need to find the magnitudes of the final and initial velocity vectors:
Final Velocity = current velocity + Δv
= (4.00i - 3.00j) m/s + (6.00i + 2.00j) m/s
= (4.00 + 6.00)i + (-3.00 + 2.00)j m/s
= 10.00i - 1.00j m/s
Initial Velocity = current velocity
= 4.00i - 3.00j m/s
Now, we can calculate the magnitudes of the final and initial velocity vectors:
Magnitude of Final Velocity = √(10.00^2 + (-1.00)^2) = √(100 + 1) = √101 ≈ 10.05 m/s
Magnitude of Initial Velocity = √(4.00^2 + (-3.00)^2) = √(16 + 9) = √25 = 5.00 m/s
Now we can calculate the change in kinetic energy:
Change in Kinetic Energy = 0.5 * mass * (final velocity^2 - initial velocity^2)
= 0.5 * 2.00 kg * ((10.05 m/s)^2 - (5.00 m/s)^2)
= 0.5 * 2.00 kg * (100.5025 m^2/s^2 - 25.00 m^2/s^2)
= 0.5 * 2.00 kg * 75.5025 m^2/s^2
= 150.00 Joules
Therefore, the net work done on the object when its velocity changes to 6.00i + 2.00j m/s is 150.00 Joules.