In a motorcycle race, one lap of the course is 650 m. At the start of the race, John sets off 4 seconds after Tom does, but John drives his motorcycle 5m/s faster and finishes the lap 2.5 seconds sooner than Tom does.

What is the speed at which each of them is driving?

What is the time taken by each of them to cover the distance.

Tom's speed = X m/s.

John's speed = (X + 5) m/s.

Tom's time = Ys.
John's time=(Y - 2.5)S.

Eq1: X*Y = 650m.
Y = 650/X.

Eq2:(X + 5)(Y - 2.5) = 650m.

In Eq2, substitute 650/X for Y:
(X + 5)(650/X - 2.5) = 650,
650 - 2.5X + 3250/X -12.5 = 650,
650 - 2.5X + 3250/X - 12.5 -650 = 0,
-2.5X + 3250/X -12.5 = 0,
Multiply each side by -X:
2.5X^2 - 3250 + 12.5X = 0,
Divide each side by 2.5:
X^2 + 5X - 1300 = 0,
Solve for X using Quad.Formula and get:
X = 33.64; X = -38.64.
Select positive value of X:
X = 33.64m/s = Tom's speed.
X*Y = 650,
33.64Y = 650,
Y = 19.3s = Tom's time.

X + 5 = 33.64 + 5 = 38.64m/s = John's
speed.
John's time = Y - 2.5 = 19.3 - 2.5 -
16.8S.

X - 20 = 0,

X + 25 =0,
X = -25.
Choose the positive solution:
X = 20m/s = Tom's speed.

X + 5 = 20 + 5 = 25m/s = John's speed.

XY = 650,
20Y = 650,
Y = 650 / 20 = 32.5s. = Tom's time.

Y -6.5 = 32.5 - 6.5=26s. = John's time.



;

OOPS!

Please disregard the 2nd (bottom)procedure.

Alternate Approach:

Tom's speed = X m/s.
Tom's time = Ys.

John's speed = (X + 5)m/s.
John's time = Y - 2.5 - 4 = Y - 6.5.

Eq1: XY = 650m. Y = 650/X.
Eq2: (X + 5)(Y - 6.5) = 6.5m.

Substitute 650/X for Y in Eq2:

(X + 5)(650/X - 6.5) = 650,
650 - 6.5X + 3250/X - 32.5 = 650,
650 - 6.5X + 3250/X -32.5 -650 = 0,
-6.5X + 3250/X -32.5 = 0,
Multiply each side by -X:
6.5X^2 - 3250 + 32.5X = 0,
Divide each term by 6.5:
X^2 + 5X - 500 = 0,
(X - 20)(X + 25) = 0,

X - 29 = 0,
X = 20.

X + 25 = 0,
X = -25.

Select positive value of X:
Tom's speed = X = 20m/s.
XY = 650,
20^Y = 650,
Tom's time = Y = 650/20 = 32.5s.

John's speed = X + 5 = 20 + 5 = 25m/s.
John's time = 32.5 - 2.5 = 30s.

To solve this problem, we can set up a system of equations based on the given information. Let's use the variable T to represent Tom's time taken to complete the lap, and J to represent John's time taken.

We know that Tom drives the lap in T seconds, so his speed can be calculated as distance divided by time:
Tom's speed (in m/s) = 650m / T seconds

John, on the other hand, finishes the lap 2.5 seconds sooner than Tom, so we can calculate his time taken as:
John's time taken (in seconds) = T - 2.5 seconds

Additionally, we are told that John drives his motorcycle 5m/s faster than Tom. Therefore, we can calculate John's speed as:
John's speed (in m/s) = Tom's speed + 5m/s

Now, let's set up the equations:

Tom's speed = John's speed - 5
650 / T = 650 / (T - 2.5) - 5

To solve this equation, we can multiply both sides by T(T-2.5) to get rid of the denominators:
650(T-2.5) = 650T - 5T(T-2.5)

Expanding and simplifying the equation further:
650T - 1625 = 650T - 5T^2 + 12.5T

Rearranging the equation:
5T^2 - 12.5T - 1625 = 0

Now, we can solve this quadratic equation for T using the quadratic formula:
T = (-b +/- sqrt(b^2 - 4ac)) / 2a

Substituting the values into the formula:
T = (-(-12.5) +/- sqrt((-12.5)^2 - 4 * 5 * -1625)) / (2 * 5)

Simplifying the equation:
T = (12.5 +/- sqrt(156.25 + 32500)) / 10
T = (12.5 +/- sqrt(32656.25)) / 10
T ≈ (12.5 +/- 180.54) / 10

We discard the negative value since time cannot be negative, so:
T ≈ (12.5 + 180.54) / 10 ≈ 19.05 seconds

Using this value of T, we can find John's time:
John's time taken ≈ T - 2.5 ≈ 19.05 - 2.5 ≈ 16.55 seconds

Finally, we can substitute the value of T into the equation for Tom's speed to find his speed:
Tom's speed ≈ 650 / 19.05 ≈ 34.13 m/s

John's speed will be 5m/s faster than Tom's speed:
John's speed ≈ 34.13 + 5 ≈ 39.13 m/s

So, Tom's speed is approximately 34.13 m/s, and John's speed is approximately 39.13 m/s.