https://file.aiquickdraw.com/mathbot/file/1729156856227cal98cgv.jpg

Let's work through the questions on the sheet:

### 23) Two objects are launched horizontally from the same height.

- **(a) Which hits the ground first?**  
  Since they are both launched horizontally from the same height, the vertical motion is independent of their horizontal velocity. They experience the same gravitational acceleration, so they will hit the ground at the same time.

- **(b) Which has the greatest range?**  
  The object with the greater horizontal velocity will cover more distance before it hits the ground. In this case, the object launched at **21 m/s** will have the greatest range compared to the one launched at **12 m/s**.

### 24) Three objects are launched horizontally from different heights and at different velocities.

- **(a) Rank the balls in order from which hits the ground first to last.**  
  The time to hit the ground depends only on the height from which they are dropped, not their horizontal velocities. From the diagram:
  - Ball 1 (4 m/s) is launched from the lowest height.
  - Ball 2 (13 m/s) is launched from a higher height.
  - Ball 3 (24 m/s) is launched from the highest height.  
  **Ranking: Ball 1 hits first, followed by Ball 2, then Ball 3.**

- **(b) If the ball traveling at 24 m/s hits the ground after 2.7 s, how high was it launched from?**  
  Using the formula for free fall:  
  \[
  h = \frac{1}{2} g t^2
  \]  
  where \( g \approx 9.8 \, \text{m/s}^2 \) and \( t = 2.7 \, \text{s} \):  
  \[
  h = \frac{1}{2} \times 9.8 \times (2.7)^2 = 35.7 \, \text{m}
  \]  
  So, it was launched from a height of approximately **35.7 meters**.

### 25) Rank the objects based on acceleration under a 50.0 N force.

Using Newton's Second Law, \( F = ma \) or \( a = \frac{F}{m} \), we can calculate the acceleration for each object:
- Object A (12.0 kg):  
  \[
  a = \frac{50}{12} \approx 4.17 \, \text{m/s}^2
  \]
- Object B (6.0 kg):  
  \[
  a = \frac{50}{6} \approx 8.33 \, \text{m/s}^2
  \]
- Object C (9.8 kg):  
  \[
  a = \frac{50}{9.8} \approx 5.10 \, \text{m/s}^2
  \]
- Object D (14.0 kg):  
  \[
  a = \frac{50}{14} \approx 3.57 \, \text{m/s}^2
  \]
  **Ranking: B > C > A > D** (greatest to least acceleration).

### 26) Rank objects with equal mass based on the applied forces.

Again, using \( a = \frac{F}{m} \), objects with greater forces will have greater accelerations. Here are the forces:
- Object A: 40 N  
- Object B: 60 N  
- Object C: 80 N  
- Object D: 100 N  
  **Ranking: D > C > B > A** (greatest to least acceleration).

Let me know if you'd like further details or explanations on any part of this!

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Here are five related questions to expand on these ideas:
1. How does changing the mass affect the time it takes an object to hit the ground if air resistance is negligible?
2. How would air resistance impact the range of an object launched horizontally?
3. What is the effect of launch angle on the range of a projectile, and why does 45 degrees provide maximum range?
4. How would doubling the force on an object affect its acceleration, assuming constant mass?
5. If two objects with different masses are pushed with the same force, why does the lighter object accelerate more?

**Tip:** When dealing with horizontal and vertical motion in projectile problems, always treat them separately. Vertical motion is influenced by gravity, while horizontal motion depends on the initial velocity and is unaffected by gravity (in the absence of air resistance).

Predict mathematically and describe how changes to initial conditions (height and horizontal velocity) affect the time of flight and range for horizontal projectiles.

Learn how to predict projectile motion and analyze how initial conditions like height and velocity affect time of flight and range. This problem involves key physics concepts such as kinematics, Newton's second law, and the independence of horizontal and vertical motion.

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