The image presents a physics problem related to the motion of a body with a mass of 2 kg, with the equation of motion given as:

$x = -2 + 4t + 2t^2$

There are various fields to complete in this problem:

1. Gravitational Force (already given as 19.6 N, which is consistent with $F = mg$, where $g = 9.8 \, \text{m/s}^2$ and $m = 2 \, \text{kg}$).

2. Initial Momentum (the equation for momentum is $p = mv$, where $v$ is the velocity).

3. Initial Kinetic Energy (calculated using $KE = \frac{1}{2} mv^2$).

4. Resultant Force (possibly derived from Newton's second law: $F = ma$, where $a$ is the acceleration).

Let's break down the missing parts:

Velocity and Acceleration:

The equation $x = -2 + 4t + 2t^2$ represents the position of the object as a function of time $t$.

• Velocity $v(t)$ is the first derivative of $x(t)$ with respect to $t$: $v(t) = \frac{dx}{dt} = 4 + 4t$ At $t = 0$, the initial velocity $v(0) = 4 \, \text{m/s}$.

• Acceleration $a(t)$ is the second derivative of $x(t)$ with respect to $t$: $a(t) = \frac{d^2x}{dt^2} = 4 \, \text{m/s}^2$

Initial Momentum:

Using $p = mv$ and $v(0) = 4 \, \text{m/s}$: $p(0) = 2 \, \text{kg} \times 4 \, \text{m/s} = 8 \, \text{kg} \cdot \text{m/s}$

Initial Kinetic Energy:

The kinetic energy is given by $KE = \frac{1}{2} mv^2$: $KE(0) = \frac{1}{2} \times 2 \, \text{kg} \times (4 \, \text{m/s})^2 = 16 \, \text{J}$

Resultant Force:

Using $F = ma$ and the constant acceleration $a = 4 \, \text{m/s}^2$: $F = 2 \, \text{kg} \times 4 \, \text{m/s}^2 = 8 \, \text{N}$

Summary of Answers:

1. Gravitational Force (N): 19.6 (given)
2. Initial Momentum (kg·m/s): 8
3. Initial Kinetic Energy (J): 16
4. Resultant Force (N): 8

Would you like more details or explanations on any part?

Here are five related questions for further exploration:

1. How does the object's velocity change with time?
2. What would the initial kinetic energy be if the mass were different?
3. How would you calculate the work done by the resultant force?
4. How does acceleration impact the object's motion over time?
5. Can you relate this to the principle of conservation of momentum?

Tip: Remember, differentiating position gives velocity, and differentiating velocity gives acceleration!