if a 10 kg weight has 10 newtons applied at 45 degrees, how to get its trajectory?

To determine the trajectory of a 10 kg weight with a 10 N force applied at a 45-degree angle, we need to follow a step-by-step process rooted in Newton's laws of motion and kinematic equations. Here's how to approach it:

### Step 1: Resolve the force into components
The force applied at an angle can be broken down into horizontal (\(F_x\)) and vertical (\(F_y\)) components using trigonometry.

- \(F_x = F \cdot \cos(\theta) = 10 \, \text{N} \cdot \cos(45^\circ) = 10 \, \text{N} \cdot \frac{\sqrt{2}}{2} \approx 7.07 \, \text{N}\)
- \(F_y = F \cdot \sin(\theta) = 10 \, \text{N} \cdot \sin(45^\circ) = 10 \, \text{N} \cdot \frac{\sqrt{2}}{2} \approx 7.07 \, \text{N}\)

### Step 2: Calculate the acceleration
Next, use Newton's second law to find the acceleration components.

- Horizontal acceleration: \(a_x = \frac{F_x}{m} = \frac{7.07 \, \text{N}}{10 \, \text{kg}} = 0.707 \, \text{m/s}^2\)
- Vertical acceleration: \(a_y = \frac{F_y}{m} - g = \frac{7.07 \, \text{N}}{10 \, \text{kg}} - 9.8 \, \text{m/s}^2 = -9.093 \, \text{m/s}^2\) (considering gravity acting downward with \(g = 9.8 \, \text{m/s}^2\))

### Step 3: Use kinematic equations
To calculate the trajectory, we use the following kinematic equations for projectile motion:

1. Horizontal motion (no horizontal acceleration besides \(a_x\)):
   - \(x(t) = v_{0x} t + \frac{1}{2} a_x t^2\)

2. Vertical motion (including gravitational acceleration):
   - \(y(t) = v_{0y} t + \frac{1}{2} a_y t^2\)

For simplicity, assume initial velocities \(v_{0x} = v_{0y} = 0\) (assuming the object is at rest before the force is applied). So the equations simplify to:

- \(x(t) = \frac{1}{2} a_x t^2 = \frac{1}{2} (0.707) t^2\)
- \(y(t) = \frac{1}{2} a_y t^2 = \frac{1}{2} (-9.093) t^2\)

### Step 4: Time of flight and trajectory shape
To find the full trajectory, you need to determine how long the object remains in motion (time of flight) and the horizontal distance it covers. You can solve the above equations for different values of time \(t\) to get the position coordinates \((x, y)\).

For example, solving for \(y(t) = 0\) will give you the time when the object hits the ground again, and you can substitute that time into \(x(t)\) to find the horizontal distance.

### Do you want a step-by-step breakdown with numerical integration, or shall I proceed further on specific aspects of this problem?

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**Questions to further explore this topic:**

1. How does gravity affect the vertical motion of the object compared to the horizontal motion?
2. What happens to the trajectory if the angle of the applied force changes?
3. How would air resistance alter the trajectory?
4. Can we use energy conservation to predict any part of the object's motion?
5. What are the differences in trajectory if the mass of the object changes but the force remains constant?

**Tip:** Always resolve forces into components when dealing with vectors at an angle.

If a 10 kg weight has 10 newtons applied at 45 degrees, how to get its trajectory?

Learn how to calculate the trajectory of a 10 kg object subjected to a 10 N force at a 45-degree angle. This step-by-step guide covers vector decomposition, Newton's laws, and kinematic equations for projectile motion. Ideal for students in Grades 10-12 studying physics and mathematics.

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