To solve the problem, we need to analyze the motion of a particle moving in a two-dimensional plane defined by the equation \( y = \frac{x^2}{2} \). We will find the relationships between the x and y components of acceleration, \( a_x \) and \( a_y \), and determine which statement is incorrect.
### Step 1: Understand the trajectory
The trajectory of the particle is given by the equation:
\[
y = \frac{x^2}{2}
\]
This represents a parabolic path in the x-y plane.
### Step 2: Find the velocity components
The velocity components in the x and y directions can be expressed as:
\[
v_x = \frac{dx}{dt}, \quad v_y = \frac{dy}{dt}
\]
Using the chain rule for differentiation, we can relate \( v_y \) to \( v_x \):
\[
v_y = \frac{dy}{dx} \cdot v_x
\]
Differentiating \( y = \frac{x^2}{2} \) with respect to \( x \):
\[
\frac{dy}{dx} = x
\]
Thus, we have:
\[
v_y = x v_x
\]
### Step 3: Find the acceleration components
The acceleration components can be found by differentiating the velocity components:
\[
a_x = \frac{dv_x}{dt}, \quad a_y = \frac{dv_y}{dt}
\]
Using the product rule for \( v_y \):
\[
a_y = \frac{d}{dt}(x v_x) = \frac{dx}{dt} v_x + x \frac{dv_x}{dt} = v_x v_x + x a_x = v_x^2 + x a_x
\]
### Step 4: Analyze the initial conditions
At \( t = 0 \), the particle starts from the origin (0, 0) with an initial speed of \( 1 \, \text{m/s} \):
\[
v_x = 1 \, \text{m/s}, \quad v_y = 0 \, \text{m/s} \quad \text{(since at } x = 0, y = 0\text{)}
\]
Thus, at the origin:
\[
a_y = v_x^2 + x a_x = 1^2 + 0 \cdot a_x = 1 \, \text{m/s}^2
\]
### Step 5: Evaluate the statements
Now we can evaluate the statements given in the problem to determine which one is incorrect based on our findings:
1. If \( a_x = 1 \, \text{m/s}^2 \), then \( a_y \) can be calculated as:
\[
a_y = 1 + 0 \cdot 1 = 1 \, \text{m/s}^2
\]
This statement is correct.
2. If \( a_x = 0 \, \text{m/s}^2 \), then:
\[
a_y = 1 + 0 \cdot 0 = 1 \, \text{m/s}^2
\]
This statement is also correct.
3. At \( t = 0 \), the particle's velocity along the x direction is \( 1 \, \text{m/s} \) and \( v_y = 0 \, \text{m/s} \). This statement is correct.
4. If \( a_x = 0 \) implies that at \( t = 1 \, \text{s} \), the angle between the particle's velocity and the x-axis is \( 45^\circ \). This statement is incorrect because if \( a_x = 0 \), then \( v_x \) remains constant, and \( v_y \) will not equal \( v_x \) at \( t = 1 \, \text{s} \).
### Conclusion
The incorrect statement is:
- If \( a_x = 0 \), then at \( t = 1 \, \text{s} \), the angle between the particle's velocity and the x-axis is \( 45^\circ \).
