Frictional Retardation in Air

When an object moves through air, it experiences a force called air resistance or drag. This force opposes the motion of the object and causes a retarding effect. The amount of air resistance depends on the shape, size, and speed of the object, as well as the density and viscosity of the air.

Vertical Retardation

If an object moves vertically upward in air, the air resistance acts opposite to the direction of motion. This means that the air resistance reduces the upward acceleration of the object, which is otherwise caused by gravity. In other words, the net upward force on the object is less than its weight, and hence the object moves with a decreasing velocity.

The magnitude of the vertical retardation due to air resistance can be calculated using the formula:

R = kv²

where R is the retardation, k is a constant that depends on the shape and size of the object, and v is the velocity of the object. For a spherical object, the value of k is approximately 0.5ρCDA/m, where ρ is the density of air, CD is the drag coefficient, A is the cross-sectional area of the object, and m is the mass of the object.

If the object moves with a constant velocity v, then the air resistance balances the weight of the object, and we have:

kv² = mg

where g is the acceleration due to gravity. Solving for v, we get:

v = sqrt(g/k)

For a spherical object, the value of k is proportional to the radius r of the sphere, and hence we have:

v = sqrt(g/kr)

Thus, the velocity of the object increases with the radius of the sphere. However, the vertical retardation also increases with the velocity, as given by the formula above.

Maximum Height

When an object is projected vertically upward, it reaches a maximum height before falling back to the ground. This maximum height can be calculated using the formula:

h = v²/2g

where h is the maximum height, v is the initial velocity of projection, and g is the acceleration due to gravity.

If the object experiences a vertical retardation due to air resistance, then its initial velocity will decrease with time, and hence its maximum height will also decrease. The percentage decrease in the maximum height can be calculated as:

delta_h/h * 100 = delta_v/v * 100

where delta_h is the decrease in the maximum height, h is the original maximum height, delta_v is the decrease in the initial velocity, and v is the original initial velocity.

For the given problem, we are given that the vertical retardation due to air resistance is 10% of the acceleration due to gravity, i.e., R = 0.1g. Hence, we have:

k = 0.5ρCDA/m = mg/v² = 10R/v² = 1/(2v²)

Substituting this value of k in the formula for v, we get:

v = sqrt(g/kr) = sqrt(2g)

Thus, the initial velocity of projection is independent of the size or shape of the object, and depends only on the acceleration due to gravity.

Now, we can calculate the original maximum height using the formula above:

h = v²/2g = 1/4g

If the object experiences a 10% decrease