Complete Question
The diagram for this question is shown on the first uploaded image
Answer:
The distance traveled in horizontal direction is ![D = 1.38 m](https://tex.z-dn.net/?f=D%20%3D%201.38%20m)
Explanation:
From the question we are told that
The length of the string is ![L = 1.6 \ m](https://tex.z-dn.net/?f=L%20%3D%201.6%20%5C%20m)
The mass of the ball is ![m = 200 g = \frac{200}{1000} = 0.2 \ kg](https://tex.z-dn.net/?f=m%20%3D%20200%20g%20%3D%20%5Cfrac%7B200%7D%7B1000%7D%20%3D%200.2%20%5C%20kg)
The height of ball is ![h = 1.5 \ m](https://tex.z-dn.net/?f=h%20%3D%201.5%20%5C%20m)
Generally the work energy theorem can be mathematically represented as
![PE = KE](https://tex.z-dn.net/?f=PE%20%3D%20KE)
Where PE is the loss in potential energy which is mathematically represented as
![PE =mgh](https://tex.z-dn.net/?f=PE%20%3Dmgh)
Where h is the difference height of ball at A and at B which is mathematically represented as
![h = y_A - y_B](https://tex.z-dn.net/?f=h%20%3D%20y_A%20-%20y_B)
So
While KE is the gain in kinetic energy which is mathematically represented as
![KE = \frac{1}{2 } (v_b ^2 - 0)](https://tex.z-dn.net/?f=KE%20%20%20%3D%20%5Cfrac%7B1%7D%7B2%20%7D%20%28v_b%20%5E2%20-%200%29)
Where
is the velocity of the of the ball
Therefore we have from above that
![PE =KE \equiv mg (y_A - y_B) = \frac{1}{2} m (v_b ^2 - 0)](https://tex.z-dn.net/?f=PE%20%3DKE%20%5Cequiv%20mg%20%28y_A%20-%20y_B%29%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20m%20%28v_b%20%5E2%20-%200%29)
Making
the subject we have
![v_b = \sqrt{2g (y_A - y_B)}](https://tex.z-dn.net/?f=v_b%20%3D%20%5Csqrt%7B2g%20%28y_A%20-%20y_B%29%7D)
substituting values
![v_b = \sqrt{2g (1.5 - 0.40)}](https://tex.z-dn.net/?f=v_b%20%3D%20%5Csqrt%7B2g%20%281.5%20-%200.40%29%7D)
![v_b = 4.6 \ m/s](https://tex.z-dn.net/?f=v_b%20%3D%204.6%20%5C%20m%2Fs)
Considering velocity of the ball when it hits the floor in terms of its vertical and horizontal component we have
![v_x = 4.6 m/s \ while \ v_y = 0 m/s](https://tex.z-dn.net/?f=v_x%20%3D%204.6%20m%2Fs%20%5C%20while%20%5C%20v_y%20%3D%200%20m%2Fs)
The time taken to travel vertically from the point the ball broke loose can be obtained using the equation of motion
![s = v_y t - \frac{1}{2} g t^2](https://tex.z-dn.net/?f=s%20%3D%20v_y%20t%20-%20%5Cfrac%7B1%7D%7B2%7D%20g%20t%5E2)
Where s is distance traveled vertically which given in the diagram as ![s = -0.4](https://tex.z-dn.net/?f=s%20%3D%20-0.4)
The negative sign is because it is moving downward
Substituting values
![-0.4 = 0 -\frac{1}{2} * 9.8 * t^2](https://tex.z-dn.net/?f=-0.4%20%3D%200%20-%5Cfrac%7B1%7D%7B2%7D%20%20%2A%209.8%20%2A%20t%5E2)
solving for t we have
![t = 0.3 \ sec](https://tex.z-dn.net/?f=t%20%3D%200.3%20%5C%20sec)
Now the distance traveled on the horizontal is mathematically evaluated as
![D = v_b * t](https://tex.z-dn.net/?f=D%20%3D%20%20v_b%20%2A%20t)
![D = 4.6 * 0.3](https://tex.z-dn.net/?f=D%20%3D%20%204.6%20%2A%200.3)
![D = 1.38 m](https://tex.z-dn.net/?f=D%20%3D%201.38%20m)