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. a. F F F
B. b. T T F
C. c. T F T
D. d. F F T

Answer

To analyze the statements regarding the relationship between kinetic energy and various parameters in projectile motion, we will evaluate each statement step by step.

### Step 1: Understand Kinetic Energy in Projectile Motion
The kinetic energy (KE) of a projectile is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the projectile and \( v \) is its velocity. The velocity can be broken down into its horizontal and vertical components.

### Step 2: Analyze 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:
\[ h = u_y t - \frac{1}{2} g t^2 \]
where \( u_y = u \sin \theta \) is the initial vertical velocity.
- The kinetic energy can be expressed in terms of vertical velocity \( v_y \):
\[ KE = \frac{1}{2} m (v_x^2 + v_y^2) \]
- Since \( v_y = u_y - gt \), we can see that as \( h \) changes, \( v_y \) changes linearly with respect to \( t \), leading to a linear relationship between KE and \( h \).
- Therefore, **Statement 1 is true**.

### Step 3: Analyze 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:
\[ x = u_x t \]
where \( u_x = u \cos \theta \) is the initial horizontal velocity.
- The horizontal component of velocity \( v_x \) remains constant, while the vertical component \( v_y \) changes due to gravity.
- The kinetic energy depends on both components of velocity:
\[ KE = \frac{1}{2} m (u_x^2 + v_y^2) \]
- Since \( v_y \) is not constant and varies with time, the relationship between KE and \( x \) is not linear but rather parabolic.
- Therefore, **Statement 2 is false**.

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

- As derived earlier, the kinetic energy can be expressed as:
\[ KE = \frac{1}{2} m (u_x^2 + (u_y - gt)^2) \]
- Expanding this gives:
\[ KE = \frac{1}{2} m \left( u^2 \cos^2 \theta + (u \sin \theta - gt)^2 \right) \]
- The term \( (u \sin \theta - gt)^2 \) is a quadratic function of time \( t \), indicating that the graph of KE versus time will indeed be a parabola.
- Therefore, **Statement 3 is true**.

### Conclusion
- **Statement 1:** True
- **Statement 2:** False
- **Statement 3:** True

Thus, the correct answer is that Statements 1 and 3 are true, while Statement 2 is false.