### Problem Statement: Constant Horizontal Force on a Gliding PuckA toy rocket engine is securely fastened to a large puck that can glide with negligible friction over a horizontal surface, taken as the xy-plane. The 3.40-kg puck has a velocity of \(1.40\hat{i}\) m/s at one instant. Eight seconds later, its velocity is \((6.00\hat{i} + 6.00\hat{j})\) m/s.(a) **Determine the Components of the Force:**Assuming the rocket engine exerts a constant horizontal force, find the components of the force.\[(\boxed{\phantom{0}} \hat{i} + \boxed{\phantom{0}} \hat{j}) \text{ N}\](b) **Calculate the Magnitude of the Force:**Find the magnitude of the force.\[\boxed{\phantom{0}} \text{ N}\]### Explanation:1. **Identify the Initial and Final Velocities:** - Initial velocity: \( \vec{v}_0 = 1.40\hat{i} \) m/s - Final velocity: \( \vec{v}_f = 6.00\hat{i} + 6.00\hat{j} \) m/s - Mass of the puck: \( m = 3.40 \) kg - Time interval: \( \Delta t = 8 \) s2. **Component Form of Newton's Second Law:** - \( \vec{F} = m \vec{a} \) - Acceleration, \( \vec{a} = \frac{\vec{v}_f - \vec{v}_0}{\Delta t} \)3. **Calculate the Components of Acceleration:** - \( a_x = \frac{v_{fx} - v_{0x}}{\Delta t} = \frac{6.00 \, \hat{i} - 1.40 \, \hat{i}}{8 \, \text{s}} \) - \( a_y = \frac{v_{fy} - v_{0y}}{\Delta t} = \frac{6.00 \, \hat{j} - 0 \, \hat{j}}{8 \, \text{s}} \

Answered: Image /qna-images/answer/00cefd21-921c-4ba8-b415-ae25d38c773a.jpg

A toy rocket engine is securely fastened to a large puck that can glide with negligible friction over a horizontal surface, taken as the xy plane. The 3.40-kg puck has a velocity of 1.40î m/s at one instant. Eight seconds later, its velocity is (6.00î + 6.0j) m/s. (a) Assuming the rocket engine exerts a constant horizontal force, find the components of the force. ]i)N (b) Find its magnitude.

A toy rocket engine is securely fastened to a large puck that can glide with negligible friction over a horizontal surface, taken as the xy plane. The 3.40-kg puck has a velocity of 1.40î m/s at one instant. Eight seconds later, its velocity is (6.00î + 6.0j) m/s. (a) Assuming the rocket engine exerts a constant horizontal force, find the components of the force. ]i)N (b) Find its magnitude.

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Newton’s Third Law of Motion

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Question

### Problem Statement: Constant Horizontal Force on a Gliding Puck

A toy rocket engine is securely fastened to a large puck that can glide with negligible friction over a horizontal surface, taken as the xy-plane. The 3.40-kg puck has a velocity of \(1.40\hat{i}\) m/s at one instant. Eight seconds later, its velocity is \((6.00\hat{i} + 6.00\hat{j})\) m/s.

(a) **Determine the Components of the Force:**
Assuming the rocket engine exerts a constant horizontal force, find the components of the force.
\[
(\boxed{\phantom{0}} \hat{i} + \boxed{\phantom{0}} \hat{j}) \text{ N}
\]

(b) **Calculate the Magnitude of the Force:**
Find the magnitude of the force.
\[
\boxed{\phantom{0}} \text{ N}
\]

### Explanation:

1. **Identify the Initial and Final Velocities:**
- Initial velocity: \( \vec{v}_0 = 1.40\hat{i} \) m/s
- Final velocity: \( \vec{v}_f = 6.00\hat{i} + 6.00\hat{j} \) m/s
- Mass of the puck: \( m = 3.40 \) kg
- Time interval: \( \Delta t = 8 \) s

2. **Component Form of Newton's Second Law:**
- \( \vec{F} = m \vec{a} \)
- Acceleration, \( \vec{a} = \frac{\vec{v}_f - \vec{v}_0}{\Delta t} \)

3. **Calculate the Components of Acceleration:**
- \( a_x = \frac{v_{fx} - v_{0x}}{\Delta t} = \frac{6.00 \, \hat{i} - 1.40 \, \hat{i}}{8 \, \text{s}} \)
- \( a_y = \frac{v_{fy} - v_{0y}}{\Delta t} = \frac{6.00 \, \hat{j} - 0 \, \hat{j}}{8 \, \text{s}} \

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