### Video Transcript

Two cars 𝐴 and 𝐵 are moving in the same direction on a straight road, where the velocity of car 𝐵 relative to car 𝐴 is 34 kilometers per hour. If car 𝐴 slowed down to a quarter of its velocity, the velocity of car 𝐵 relative to car 𝐴 would be 76 kilometers per hour. Determine the actual speeds of the two cars 𝑉 sub 𝐴 and 𝑉 sub 𝐵.

In this question, we are dealing with relative velocities. And we know that the velocity of car 𝐵 relative to car 𝐴 is equal to the velocity of car 𝐵 minus the velocity of car 𝐴. This formula holds for any two bodies moving along the same one-dimensional axis. In this question, we are told this is equal to 34 kilometers per hour, giving us the equation 𝑉 sub 𝐵 minus 𝑉 sub 𝐴 is equal to 34. We are also told that if car 𝐴 slows down to a quarter of its velocity, the relative velocity is 76 kilometers per hour. This means that 𝑉 sub 𝐵 minus a quarter of 𝑉 sub 𝐴 is equal to 76. We now have a pair of simultaneous equations that we can solve to calculate 𝑉 sub 𝐴 and 𝑉 sub 𝐵.

We can solve these equations by elimination or substitution. In this question, we will rearrange equation one first. Adding 𝑉 sub 𝐴 to both sides of this equation, we have 𝑉 sub 𝐵 is equal to 𝑉 sub 𝐴 plus 34. We can now substitute this into equation two. This gives us 𝑉 sub 𝐴 plus 34 minus a quarter 𝑉 sub 𝐴 is equal to 76. Collecting like terms on the left-hand side and subtracting 34 from both sides, we have three-quarters 𝑉 sub 𝐴 is equal to 42. We can then divide through by three-quarters such that 𝑉 sub 𝐴 is equal to 56. The speed of car 𝐴 is 56 kilometers per hour.

We can then substitute this value back in to equation one. We have 𝑉 sub 𝐵 minus 56 is equal to 34. Adding 56 to both sides of this equation gives us 𝑉 sub 𝐵 is equal to 90. The speed of car 𝐵 is 90 kilometers per hour. We can therefore conclude that the actual speeds of the two cars 𝐴 and 𝐵 are 56 and 90 kilometers per hour, respectively.