A motorcyclist moving with uniform retardation takes 10s and 20s to travel successive quarter kilometres. How much further will he travel before coming to rest?

To find out how much further the motorcyclist will travel before coming to rest, you can use the equations of motion for uniformly retarded motion. The key information you have is that it takes 10 seconds to travel the first quarter-kilometer and 20 seconds to travel the next quarter-kilometer.

Let's denote the initial velocity as "u" (which we want to find), the final velocity as "v" (which is 0 since the motorcyclist comes to rest), the time taken as "t," and the distance as "s."

For the first quarter-kilometer (0.25 km):

Initial velocity, u = ?
Final velocity, v = 0 m/s
Time taken, t = 10 seconds
Distance, s = 0.25 km = 250 meters
You can use the equation of motion:

s = ut + (1/2)at^2

Where:

s is the distance traveled,
u is the initial velocity,
t is the time taken, and
a is the acceleration (which is negative since it's retardation in this case).
Let's calculate the acceleration for the first quarter-kilometer:

s = ut + (1/2)at^2
250 = u(10) + (1/2)a(10^2)
250 = 10u + 5a

Now, for the second quarter-kilometer (0.25 km):

Initial velocity, u = 0 m/s (since the motorcyclist starts from rest)
Final velocity, v = 0 m/s
Time taken, t = 20 seconds
Distance, s = 0.25 km = 250 meters
Using the same equation of motion:

s = ut + (1/2)at^2
250 = 0(20) + (1/2)a(20^2)
250 = 20a(2)

Now, you have two equations for acceleration (a) from the first and second quarters:

250 = 10u + 5a
250 = 40a
Solve these two equations simultaneously to find the values of u and a:

From equation 2:
a = 250 / 40
a = 6.25 m/s²

Now, substitute the value of a into equation 1:

250 = 10u + 5(6.25)
250 = 10u + 31.25

Subtract 31.25 from both sides:

10u = 250 - 31.25
10u = 218.75

Now, divide by 10:

u = 218.75 / 10
u = 21.875 m/s

So, the initial velocity of the motorcyclist is approximately 21.875 m/s.

To find how much further the motorcyclist will travel before coming to rest, you can use the equation of motion with v = 0:

0 = (21.875)m + (1/2)(-6.25)(t^2)

Solve for t:

6.25(t^2) = 21.875
t^2 = 21.875 / 6.25
t^2 = 3.5

Now, find t:

t = √(3.5)
t ≈ 1.87 seconds

The motorcyclist will travel for approximately 1.87 seconds before coming to rest. To find the distance traveled during this time, use the equation of motion:

s = ut + (1/2)at^2
s = (21.875)(1.87) + (1/2)(-6.25)(1.87^2)

Calculate s:

s ≈ 40.74 meters

So, the motorcyclist will travel approximately 40.74 meters further before coming to rest.