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.

Question

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.

Answer 1

RT = D

T = time for Tom
T - 6.5 = time for John (-4s + -2.5s)
R = rate for Tom
R + 5 = rate for John
D = 650

RT = D
RT = 650
(R + 5)(T - 6.5) = 650

Solve these equations simultaneously.
Expand (R + 5)(T - 6.5) = 650
Then substitute, RT = 650 for RT in the expansion.

Answer 2

Why did you add -4s + -2.5s ?

Answer 3

John sets off 4 seconds after Tom does, and finishes the lap 2.5 seconds sooner than Tom does.

So, John's total time will be 6.5 sec less than Tom's time.

Do you understand?

Also, If you solved correctly, R = 20 and T = 32.5.

Remember, the problem wants each racers' speed and time.
So use the below to calculate that after you solve the equations for T and R.

T = time for Tom
R = rate for Tom
T - 6.5 = time for John
R + 5 = rate for John

Answer 4

Okay, I understand now, thanks!

Answer 5

You're welcome!

Answer 6

To solve this problem, we can set up a system of equations based on the given information. Let's assign variables to the unknown quantities:

Let's assume that Tom's speed is 't' meters per second and John's speed is 'j' meters per second.

We know that the length of the lap is 650 m. So, the time taken by Tom to complete the lap is given by:

Time taken by Tom = Distance / Speed = 650 / t seconds -- Equation 1

John drives his motorcycle 5 m/s faster than Tom, so his speed is t + 5 m/s. He finishes the lap 2.5 seconds sooner than Tom, so the time taken by John to complete the lap is given by:

Time taken by John = Distance / Speed = 650 / (t + 5) seconds -- Equation 2

It is also given that John sets off 4 seconds after Tom does. Therefore, the race duration for John is 4 seconds less than that of Tom:

Time taken by John = Time taken by Tom - 4 seconds

Substituting the values from Equations 1 and 2:

650 / (t + 5) = 650 / t - 4 -- Equation 3

Now, we can solve this equation to find the values of t and j.

To solve Equation 3, we can cross-multiply:

650t = 650(t + 5) - 4t(t + 5)

Simplifying the equation further:

650t = 650t + 3250 - 4t^2 - 20t

Rearranging the terms:

4t^2 + 20t - 3250 = 0

Dividing both sides of the equation by 2:

2t^2 + 10t - 1625 = 0

Now, we can solve this quadratic equation to find the values of t and eventually j.

Using the quadratic formula:

t = (-10 ± √(10^2 - 4(2)(-1625))) / (2(2))

Simplifying further:

t = (-10 ± √(10000 + 13000)) / 4

t = (-10 ± √23000) / 4

Now, calculating the values of t:

t = (-10 + √23000) / 4 ≈ 28.21 m/s (rounded to two decimal places)

t = (-10 - √23000) / 4 ≈ -38.71 m/s (rounded to two decimal places)

Since speed cannot be negative, we disregard the negative value.

Therefore, Tom's speed, t, is approximately 28.21 m/s.

To find John's speed, j, we add 5 m/s to Tom's speed:

j ≈ t + 5 ≈ 28.21 + 5 ≈ 33.21 m/s (rounded to two decimal places)

Hence, Tom's speed is approximately 28.21 m/s, and John's speed is approximately 33.21 m/s.

To find the time taken by each of them to cover the distance:

Time taken by Tom = 650 / t ≈ 650 / 28.21 ≈ 23.04 seconds (rounded to two decimal places)

Time taken by John = 650 / j ≈ 650 / 33.21 ≈ 19.58 seconds (rounded to two decimal places)

Therefore, Tom takes approximately 23.04 seconds to cover the distance, while John takes approximately 19.58 seconds.