A cannon on a level plane is aimed at an angle `alpha` above the horizontal and a shell is fired with a muzzle velocity `v_(0)` towards a vertical cliff a distance R away. Then the height from the bottom at which the shell strikes the side walls of the cliff is

Options

A. (a)` R sin alpha -1/2 (gR^2)/(v_0^2 sin^2alpha)`
B. (b)`R cos alpha- 1/2 (gR^2)/(v_0^2cos ^2 alpha)`
C. (c)`R tan alpha -1/2 (gR^2)/(v_0^2 sin ^2 alpha)`
D. (d)`R tan alpha -1/2 (gR^2)/(v_0^2 cos ^2 alpha)`

Explanation

To solve the problem of determining the height at which the shell strikes the vertical cliff, we can break down the motion of the shell into horizontal and vertical components. Here’s a step-by-step solution:

### Step 1: Identify the initial conditions
- The cannon fires a shell with an initial velocity \( v_0 \) at an angle \( \alpha \) above the horizontal.
- The horizontal distance to the cliff is \( R \).

### Step 2: Resolve the initial velocity into components
- The horizontal component of the initial velocity \( v_{0x} \) is given by:
\[ v_{0x} = v_0 \cos \alpha \]
- The vertical component of the initial velocity \( v_{0y} \) is given by:
\[ v_{0y} = v_0 \sin \alpha \]

### Step 3: Determine the time of flight to reach the cliff
- The time \( t \) taken to reach the cliff can be found using the horizontal motion equation:
\[ R = v_{0x} \cdot t \implies t = \frac{R}{v_0 \cos \alpha} \]

### Step 4: Calculate the vertical position when the shell reaches the cliff
- The vertical position \( y \) of the shell at time \( t \) can be calculated using the vertical motion equation:
\[ y = v_{0y} \cdot t - \frac{1}{2} g t^2 \]
- Substituting \( v_{0y} \) and \( t \):
\[ y = (v_0 \sin \alpha) \left(\frac{R}{v_0 \cos \alpha}\right) - \frac{1}{2} g \left(\frac{R}{v_0 \cos \alpha}\right)^2 \]

### Step 5: Simplify the equation
- The first term simplifies to:
\[ y = R \tan \alpha \]
- The second term simplifies to:
\[ \frac{1}{2} g \left(\frac{R^2}{v_0^2 \cos^2 \alpha}\right) \]
- Thus, the equation for \( y \) becomes:
\[ y = R \tan \alpha - \frac{g R^2}{2 v_0^2 \cos^2 \alpha} \]

### Final Result
- The height \( h \) at which the shell strikes the cliff is:
\[ h = R \tan \alpha - \frac{g R^2}{2 v_0^2 \cos^2 \alpha} \]