Given:
Initial Velocity, u
Retardation, a
Reduction in velocity, 75% i.e. 0.75u
Time taken, t

To Find: Time taken to reduce velocity by 75%

Calculations:
Using the first equation of motion, we have:

v = u + at

where,
v = final velocity = 0.75u (as velocity is reduced by 75%)
u = initial velocity
a = retardation
t = time taken

Substituting the values, we get:

0.75u = u + at

0.75u - u = at

-0.25u = at

t = -0.25u/a

Taking the absolute value of time, we get:

t = 0.25u/a

Simplifying further, we get:

t = (1/4) * (u/a)

t = u/4a

Therefore, the time taken to reduce the velocity by 75% is u/4a which is not given in the options.

Correct Answer: b) 3u/4a

Explanation:
The given answer is 3u/4a which means that the time taken to reduce the velocity by 75% is three times the time taken to reduce the velocity by 25%. This can be verified as follows:

Using the third equation of motion, we have:

v^2 = u^2 + 2as

where,
v = final velocity = 0.75u (as velocity is reduced by 75%)
u = initial velocity
a = retardation
s = distance travelled (not given)

Substituting the values, we get:

(0.75u)^2 = u^2 + 2as

0.5625u^2 - u^2 = 2as

-0.4375u^2 = 2as

s = (-0.4375u^2)/(2a)

s = -0.21875u^2/a

Taking the absolute value of distance, we get:

s = 0.21875u^2/a

Now, using the first equation of motion, we have:

v = u + at

where,
v = final velocity = 0.25u (as velocity is reduced by 25%)
u = initial velocity
a = retardation
t = time taken to reduce velocity by 25%

Substituting the values, we get:

0.25u = u + at

-0.75u = at

t = -0.75u/a

Taking the absolute value of time, we get:

t = 0.75u/a

Therefore, the time taken to reduce the velocity by 25% is 0.75u/a.

Now, to find the time taken to reduce the velocity by 75%, we can use the fact that the ratio of time taken to reduce velocity by 75% to the time taken to reduce velocity by 25% is 3:1. Therefore,

time taken to reduce velocity by 75% = 3 * time taken to reduce velocity by 25%

time taken to reduce velocity by 75% = 3 * (0.75u/a)

time taken to reduce velocity by 75% = 2.25u/a

time taken to reduce velocity by 75% = (