a ball of mass 2kg is suspended by a string of length 2m.it then from "b" and kicks an identical ball at position "a".if frictional force is neglected,through what distance ball "a"is moved horizontal if ball"b"becomes at rest after it kicks ball"a"?take height of the table is 3m.
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To determine the distance through which ball "a" moves horizontally, we need to calculate the potential energy lost by ball "b" and equate it to the kinetic energy gained by ball "a". Here's how we can do that:
1. Determine the initial potential energy of ball "b":
- The initial potential energy (PE) of ball "b" is given by PE = mgh, where m is the mass of the ball, g is the acceleration due to gravity, and h is the height from which it is dropped.
- In this case, the mass of both balls is 2 kg, and the height is 3 m. So, the initial potential energy of ball "b" is PE = 2 kg x 9.8 m/s^2 x 3 m = 58.8 J.
2. Determine the final potential energy of ball "b":
- Since ball "b" comes to rest after kicking ball "a," its final potential energy is 0 J.
3. Determine the kinetic energy gained by ball "a":
- The kinetic energy (KE) gained by ball "a" is given by KE = (1/2)mv^2, where m is the mass of the ball and v is its velocity.
- Since both balls are identical, the mass of ball "a" is also 2 kg, and its velocity is the same as the velocity of ball "b" just before the kick. We'll represent this velocity as v.
- Therefore, the kinetic energy gained by ball "a" is KE = (1/2) x 2 kg x v^2 = v^2 J.
4. Equate the potential energy lost by ball "b" to the kinetic energy gained by ball "a":
- Set the initial potential energy of ball "b" equal to the final potential energy of ball "b" plus the kinetic energy gained by ball "a":
PE = KE
58.8 J = 0 J + v^2 J
5. Solve for the velocity of ball "a":
- Rearrange the equation to solve for v:
v^2 = 58.8 J
v = √(58.8 J) ≈ 7.67 m/s
6. Calculate the distance ball "a" moves horizontally:
- The horizontal distance covered by ball "a" can be determined using the formula d = vt, where v is the velocity of ball "a" and t is the time it takes for ball "b" to reach the table.
- The time it takes for ball "b" to reach the table can be determined using the equation of motion, h = (1/2)gt^2, where h is the height of the table and g is the acceleration due to gravity.
- In this case, the height of the table is 3 m, so we can solve for t as follows:
3 m = (1/2) x 9.8 m/s^2 x t^2
t^2 = 3 m / (4.9 m/s^2)
t = √(3 m / (4.9 m/s^2)) ≈ 0.78 s
- Now, use the formula d = vt to calculate the distance:
d = 7.67 m/s x 0.78 s ≈ 5.99 m
Therefore, ball "a" moves horizontally for approximately 5.99 meters.