Statement -1 : The graph between - Kinetic energy and vertical displacement is a stright line for a projectile.
Statement -2 : The graph between kinetic energy and horizontal displacement is a straight is straight line for a projectile.
Statement -3 : the graph between kinetic energy and time is a parabola for a projectile.

Options

A. F F F
B. T T F
C. T F T
D. F F T

Answer

To analyze the statements regarding the graphs of kinetic energy (KE) in projectile motion, we will evaluate each statement step by step.

### Step 1: Understanding Kinetic Energy in Projectile Motion
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the projectile and \( v \) is its velocity. In projectile motion, the velocity has both horizontal and vertical components.

### Step 2: Analyzing Statement 1
**Statement 1:** The graph between kinetic energy and vertical displacement is a straight line for a projectile.

- The vertical displacement \( h \) can be expressed as:
\[ h = u \sin \theta \cdot t - \frac{1}{2} g t^2 \]
- The vertical component of velocity \( v_y \) is given by:
\[ v_y = u \sin \theta - gt \]
- The total velocity \( v \) can be expressed as:
\[ v^2 = (u \cos \theta)^2 + (u \sin \theta - gt)^2 \]
- Substituting \( v \) into the kinetic energy formula, we can see that as vertical displacement changes, the kinetic energy will change linearly with respect to vertical displacement because the vertical velocity changes linearly with time.

**Conclusion for Statement 1:** True.

### Step 3: Analyzing Statement 2
**Statement 2:** The graph between kinetic energy and horizontal displacement is a straight line for a projectile.

- The horizontal displacement \( x \) is given by:
\[ x = u \cos \theta \cdot t \]
- The horizontal component of velocity \( v_x \) remains constant:
\[ v_x = u \cos \theta \]
- As time progresses, the vertical component of velocity changes due to gravity, which means that the kinetic energy will not change linearly with horizontal displacement. Instead, it will vary as a function of both horizontal and vertical components, leading to a non-linear relationship.

**Conclusion for Statement 2:** False.

### Step 4: Analyzing Statement 3
**Statement 3:** The graph between kinetic energy and time is a parabola for a projectile.

- As time progresses, the vertical component of velocity changes, which affects the kinetic energy. The relationship between kinetic energy and time can be derived from:
\[ KE = \frac{1}{2} m (u^2 \cos^2 \theta + (u \sin \theta - gt)^2) \]
- This expression is quadratic in terms of time \( t \) because of the \( -gt \) term, leading to a parabolic relationship.

**Conclusion for Statement 3:** True.

### Final Summary of Statements
- **Statement 1:** True
- **Statement 2:** False
- **Statement 3:** True