Decrease in Maximum Height Due to Air Friction

The maximum height attained by a projectile is given by the formula:

H = u^2 sin^2θ / 2g

where u is the initial velocity, θ is the angle of projection, and g is the acceleration due to gravity.

Due to air friction, the vertical component of velocity of the projectile decreases with time. Let's assume that the vertical retardation due to air friction is 10% of the acceleration due to gravity, i.e., 0.1g.

The equations of motion for the vertical motion of the projectile are:

v = u - gt (1)

s = ut - 0.5gt^2 (2)

where v is the final velocity, s is the displacement, and t is the time taken.

Using equation (1), we can find the time taken for the projectile to reach the maximum height:

v = 0 at the maximum height

0 = u - gt

t = u/g

Now, using equation (2), we can find the maximum height H:

H = u^2 sin^2θ / 2g - 0.5g(u/g)^2

H = u^2 sin^2θ / 2g - 0.5u^2

H = u^2 (sin^2θ / 2g - 0.5)

Due to air friction, the maximum height will be decreased by an amount equal to the work done against air friction. The work done is given by:

W = Fd = mad

where F is the force of air friction, m is the mass of the projectile, and d is the distance traveled against air friction.

Since the projectile moves vertically upwards, the distance traveled against air friction is equal to the maximum height H.

The force of air friction is given by:

F = bv

where b is a constant that depends on the shape and size of the projectile, and v is the velocity of the projectile relative to the air.

At the maximum height, the velocity of the projectile is zero. Therefore, the work done against air friction is:

W = bvH

Substituting v = u/2 (since the vertical retardation due to air friction is 10% of g), we get:

W = buH/2

The decrease in maximum height is equal to the work done against air friction divided by the potential energy at the maximum height:

ΔH / H = W / (mu^2 / 2)

ΔH / H = buH / mu^2

ΔH / H = b / 2u

Substituting u = v / sinθ, we get:

ΔH / H = b sinθ / 2v

Since sinθ is less than or equal to 1, the decrease in maximum height is less than or equal to 50% of the initial maximum height.

Therefore, the correct answer is option (c) 10%.

Time Taken to Reach Maximum Height

The time taken to reach the maximum height is given by:

t = u/g

Since the initial velocity u is not affected by air friction, the time taken to reach the maximum height is not affected by air friction.

Therefore, the correct answer is option (d) None of these.