what is the kinetic energy at this time?find the total work done on the object if its velocity changes (8i+4j)

Question

A 3.00 kg abject has a velocity(6.00i-2.00j)m/s.

Answer 1

Kinetic energy is m v^2 / 2.
To find v, you need to find the magnitude of the vector that represents the object's velocity. (Actually, the above expression requires the SQUARE of the velocity, which will save us a step in the solution.)

Using vector algebra and the Pythagorean theorem, the velocity is sqrt (6^2 + 2^2). Don't bother to take the square root and you'll have v^2, which you can use in the equation for kinetic energy.

The velocity changes by 8i+4j. Let's assume that this change occurs over a period of 1 second. (It turns out that the amount of work done does not depend on the time period, so this assumption doesn't affect the answer that we will calculate.)

To change the velocity by 8i+4j, we need to apply a force of 24 newtons (8 x 3) in the i direction, and 12 newtons (4 x 3) in the j direction for 1 second.

The amount of work done by each of these forces (considered separately) is equal to the force times the distance over which it acts. The 24 newton force in the i direction acts over a distance of 10 meters (since the average i-direction velocity during this 1-second period is 10 m/s; you calculate that by averaging the initial and final velocities, 6 m/s and (6+8) m/s). So the work done is 240 newton-meters, or 240 joules.

In the j direction, the force of 12 newtons acts over a distance of 1 meter, for total work done of 12 joules. (Note that the average velocity in the j direction is 1 meter/second, found by averaging -2 m/s and (-2+4) m/s.)

So the total work done is 252 joules. (I'm a little uncomfortable about treating these separately and adding them up. If I had a physics book here, I'd read it to decide whether this is the right treatment. If you have a physics book, I suggest you do that.)

P.S. I have assumed that the velocity changes BY an amount of (8i+4j). If instead it changes TO (8i+4j), then you need to figure out the amount BY which the velocity has changed, and plus those i and j values in where I used 8 and 4.

Answer 2

Not quite clear! if velocity U=6i-2j changes & becomes V=8i+4j, then:
E2=m*(V*V)/2=m*(Vx^2+Vy^2)/2=3*(64+16)/2=120Joules
A=E2-E1, E1=m*(U*U)/2=3*(6*6+(-2)*(-2))/2=60Joules, thus
A=120-60=60