### Video Transcript

A particle of mass 150 grams was projected at 13 metres per second across a horizontal plane. It decelerated uniformly at two metres per second squared. Find the change in its kinetic energy in the first four seconds of motion.

Weโre told in this statement that the particle has a mass of 150 grams. Weโll call that mass ๐. Weโre also told that it has an initial speed of 13 metres per second. Weโll call the ๐ฃ sub ๐. The particle undergoes a deceleration of two metres per second squared. Weโll call that value ๐. We want to find the change in the kinetic energy of the particle over the first four seconds of motion. We can call this change ฮKE.

To begin our solution, letโs recall that the kinetic energy of an object is equal to one-half the mass of that object times its speed squared. Since the change in kinetic energy of our particle ฮKE is equal to its final kinetic energy minus its initial kinetic energy, the equation for kinetic energy tells us we can write this as one-half ๐๐ฃ๐ squared, where ๐ฃ๐ is the particleโs final speed, minus one-half ๐๐ฃ๐ squared or factoring out in ๐ over two one-half ๐ times ๐ฃ๐ squared minus ๐ฃ๐ squared.

We know the particleโs mass ๐ as well as its initial speed ๐ฃ sub ๐. But we want to solve for ๐ฃ sub ๐. To solve for ๐ฃ sub ๐, we can use the fact that acceleration is equal to the change in speed over the change in time. In our case, we can write that ๐ is equal to ๐ฃ sub ๐ minus ๐ฃ sub ๐, the final speed of the particle, divided by the time over which the particle decelerates, which is given in the problem statement as four seconds.

Rearranging this equation to solve for ๐ฃ sub ๐, we see that ๐ฃ sub ๐ is equal to ๐ฃ sub ๐ minus ๐ times ๐ก or plugging in for ๐ฃ sub ๐, ๐, and ๐ก, we calculate ๐ฃ sub ๐ to be five metres per second. Thatโs the speed of the particle after four seconds have passed.

We can now return to our ฮKE equation and weโre now prepared to plug in for each of the variables in this expression. When we do, being careful to write our mass in units of kilograms to be consistent with the units in the rest of our expression, we enter these values on our calculator and find that ฮKE is equal to negative 10.8 joules. Thatโs the change in kinetic energy this particle experiences.