A body is projected horizontally from the top of a tower with a velocity of 20 m/s. After what time the vertical component of velocity is four times the horizontal component of velocity ? (g = `10 m//s^(2)`)

Options

A. 16 sec
B. 8 sec
C. 4 sec
D. 2 sec

Answer

To solve the problem step by step, we will analyze the motion of the body projected horizontally from the top of a tower.

### Step 1: Understand the initial conditions
- The body is projected horizontally with an initial horizontal velocity \( u_h = 20 \, \text{m/s} \).
- The initial vertical velocity \( u_v = 0 \, \text{m/s} \) (since it is projected horizontally).
- The acceleration due to gravity \( g = 10 \, \text{m/s}^2 \) acts downward.

**Hint:** Identify the components of the initial velocity and the forces acting on the body.

### Step 2: Write the equation for vertical velocity
Using the kinematic equation for vertical motion:
\[ v_v = u_v + g \cdot t \]
Since \( u_v = 0 \):
\[ v_v = g \cdot t = 10 \cdot t \]

**Hint:** Remember that the vertical component of velocity increases due to gravity.

### Step 3: Determine the horizontal component of velocity
The horizontal component of velocity remains constant because there is no horizontal acceleration:
\[ v_h = u_h = 20 \, \text{m/s} \]

**Hint:** In projectile motion, horizontal velocity does not change unless acted upon by an external force.

### Step 4: Set up the equation based on the problem statement
According to the problem, the vertical component of velocity is four times the horizontal component of velocity:
\[ v_v = 4 \cdot v_h \]
Substituting the known values:
\[ 10 \cdot t = 4 \cdot 20 \]
\[ 10 \cdot t = 80 \]

**Hint:** Use the relationship given in the problem to set up an equation.

### Step 5: Solve for time \( t \)
Now, solve for \( t \):
\[ t = \frac{80}{10} = 8 \, \text{s} \]

**Hint:** Isolate \( t \) to find the time it takes for the vertical velocity to reach the specified condition.

### Conclusion
The time after which the vertical component of velocity is four times the horizontal component of velocity is \( t = 8 \, \text{s} \).

**Final Answer:** \( t = 8 \, \text{s} \)

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