Key Concepts and Formulas

• Uniformly Accelerated Motion: The displacement $s$ of an object with initial velocity $u$, constant acceleration $a$, after time $t$ is given by $s = ut + \frac{1}{2}at^2$.
• Uniform Motion: The displacement $s$ of an object moving with constant velocity $v$ after time $t$ is given by $s = vt$.
• The condition for the lizard catching the insect is that their positions are the same at a certain time $t$.

Step-by-Step Solution

Step 1: Define the Coordinate System and Initial Conditions

We establish a coordinate system to describe the motion of both the lizard and the insect. Let the initial position of the lizard be the origin ($x = 0$ at $t = 0$). The positive direction is the direction of motion.

• Lizard (L):
  • Initial position: $x_{L,0} = 0$ cm
  • Initial velocity: $u_L = 0$ cm/s
  • Acceleration: $a_L = 2$ cm/s$^2$
• Insect (I):
  • Initial position: Since the lizard is 21 cm behind the insect, the insect's initial position is $x_{I,0} = 21$ cm.
  • Velocity: $v_I = 20$ cm/s
  • Acceleration: $a_I = 0$ cm/s$^2$

Our goal is to find the time $t$ when $x_L(t) = x_I(t)$.

Step 2: Determine the Position of the Lizard as a Function of Time

Using the kinematic equation for uniformly accelerated motion, we have: $x_L(t) = x_{L,0} + u_L t + \frac{1}{2} a_L t^2$ Substituting the lizard's initial conditions, we get: $x_L(t) = 0 + (0)t + \frac{1}{2}(2)t^2 = t^2$ So, $x_L(t) = t^2$ cm.

Step 3: Determine the Position of the Insect as a Function of Time

Using the kinematic equation for uniform motion, we have: $x_I(t) = x_{I,0} + v_I t$ Substituting the insect's initial conditions, we get: $x_I(t) = 21 + 20t$ So, $x_I(t) = 21 + 20t$ cm.

Step 4: Solve for the Time When the Lizard Catches the Insect

The lizard catches the insect when their positions are equal, i.e., $x_L(t) = x_I(t)$. Therefore, $t^2 = 21 + 20t$ Rearranging the equation, we get a quadratic equation: $t^2 - 20t - 21 = 0$ Factoring the quadratic equation: $(t - 21)(t + 1) = 0$ This gives us two possible solutions for $t$: $t = 21 \text{ s} \quad \text{or} \quad t = -1 \text{ s}$

Step 5: Interpret the Results and Determine the Valid Solution

Since time cannot be negative, we discard the solution $t = -1$ s. The only physically meaningful solution is $t = 21$ s.

Therefore, the lizard catches the insect after 21 seconds.

Common Mistakes & Tips

• Always use a consistent coordinate system for both objects. Defining separate origins or directions can lead to incorrect equations.
• Pay close attention to the initial conditions, especially whether an object starts from rest or has an initial velocity.
• Remember to interpret your mathematical results in the context of the physical problem. Discard any solutions that are not physically possible (e.g., negative time).

Summary

We set up a coordinate system and defined the initial conditions for both the lizard and the insect. We then derived equations for their positions as functions of time using kinematic equations. By equating these positions, we obtained a quadratic equation, which we solved to find the time when the lizard catches the insect. After discarding the negative time solution, we found that the lizard catches the insect after 21 seconds.

The final answer is $\boxed{21}$, which corresponds to option (C).