Introduction:
When a ball is dropped from a height and collides with the ground, the collision is not perfectly elastic. The coefficient of restitution (COR) is a measure of how much kinetic energy is conserved during the collision. In this scenario, we are given a ball of mass 1 kg dropped from a height of 20m, and the coefficient of restitution is 0.9. We need to calculate the impulse imparted to the ground during the collision.

Formula:
The impulse imparted to an object can be calculated using the formula:

Impulse = Change in momentum = Mass * Change in velocity

Calculating the initial velocity:
When the ball is dropped, it has a potential energy due to its height. We can calculate the initial velocity using the formula:

Potential Energy = Mass * Gravity * Height
Potential Energy = 1 kg * 9.8 m/s^2 * 20 m
Potential Energy = 196 J

Since the potential energy is converted to kinetic energy when the ball falls, we can equate it to the kinetic energy formula:

Kinetic Energy = (1/2) * Mass * Velocity^2
196 J = (1/2) * 1 kg * Velocity^2
Velocity^2 = 392 m^2/s^2
Velocity = √392 m/s
Velocity = 19.8 m/s (approximately)

Calculating the final velocity:
The coefficient of restitution (COR) is the ratio of the final velocity to the initial velocity after a collision. In this case, the COR is given as 0.9.

COR = Final Velocity / Initial Velocity
0.9 = Final Velocity / 19.8 m/s
Final Velocity = 0.9 * 19.8 m/s
Final Velocity = 17.82 m/s (approximately)

Calculating the change in velocity:
The change in velocity during the collision can be calculated by subtracting the final velocity from the initial velocity.

Change in Velocity = Final Velocity - Initial Velocity
Change in Velocity = 17.82 m/s - 19.8 m/s
Change in Velocity = -1.98 m/s (approximately)

Calculating the impulse:
Now that we have the change in velocity, we can calculate the impulse imparted to the ground using the formula:

Impulse = Mass * Change in Velocity
Impulse = 1 kg * -1.98 m/s
Impulse = -1.98 N·s

Conclusion:
The impulse imparted to the ground when a ball of mass 1 kg is dropped from a height of 20m with a coefficient of restitution of 0.9 is approximately -1.98 N·s. The negative sign indicates that the impulse is in the opposite direction of the initial velocity, as the ball bounces back.