A ball is dropped on to a horizontal plate from a height h = 9 m above it. If the coefficient of restitution is e=1/2, the total distance travelled before the ball comes to rest is

Options

A. 10 m
B. 15 m
C. 20 m
D. 25 m

Answer

To solve the problem of a ball dropped from a height of \( h = 9 \, \text{m} \) with a coefficient of restitution \( e = \frac{1}{2} \), we need to determine the total distance traveled by the ball before it comes to rest. Here’s a step-by-step solution:

### Step 1: Understand the Coefficient of Restitution
The coefficient of restitution \( e \) is defined as the ratio of the speed after the collision to the speed before the collision. In this case, since \( e = \frac{1}{2} \), the ball will rebound to half the height from which it fell.

### Step 2: Calculate the First Rebound Height
When the ball is dropped from a height \( h = 9 \, \text{m} \), it will hit the ground and rebound to a height \( h_1 \):
\[ h_1 = e \cdot h = \frac{1}{2} \cdot 9 = 4.5 \, \text{m} \]

### Step 3: Calculate the Second Rebound Height
After reaching \( h_1 \), the ball will again fall and rebound to a height \( h_2 \):
\[ h_2 = e \cdot h_1 = \frac{1}{2} \cdot 4.5 = 2.25 \, \text{m} \]

### Step 4: Calculate the Subsequent Rebound Heights
Continuing this process, we can find the heights for the next rebounds:
- \( h_3 = e \cdot h_2 = \frac{1}{2} \cdot 2.25 = 1.125 \, \text{m} \)
- \( h_4 = e \cdot h_3 = \frac{1}{2} \cdot 1.125 = 0.5625 \, \text{m} \)

### Step 5: Total Distance Calculation
The total distance \( D \) traveled by the ball can be calculated as follows:
\[ D = h + h_1 + h_1 + h_2 + h_2 + h_3 + h_3 + h_4 + h_4 \]
This can be simplified to:
\[ D = h + 2(h_1 + h_2 + h_3 + h_4) \]

### Step 6: Sum of the Heights
Using the formula for the sum of a geometric series, we can express the total distance:
\[ D = h + 2 \left( h_1 + h_2 + h_3 + h_4 + \ldots \right) \]
The heights form a geometric series where:
- First term \( a = h_1 = 4.5 \)
- Common ratio \( r = \frac{1}{2} \)

The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} = \frac{4.5}{1 - \frac{1}{2}} = \frac{4.5}{\frac{1}{2}} = 9 \]

### Step 7: Final Calculation of Total Distance
Now substituting back into the total distance formula:
\[ D = 9 + 2 \cdot 9 = 9 + 18 = 27 \, \text{m} \]

### Conclusion
The total distance traveled before the ball comes to rest is:
\[ \boxed{27 \, \text{m}} \]