Example: Lemon drop A lemon drops from a tree and falls to the ground 3.15 m below. A) How much time does it take for the lemon to reach the ground? B) What is the lemon's speed right before it hits the ground?

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UpStudy AI · Accepted Answer

Sure, let's solve the problem step by step.

**Given:**
- Height from which the lemon drops, $$ h = 3.15 $$ meters
- Acceleration due to gravity, $$ g = 9.8 $$ meters per second squared ($$ m/s² $$)
- Initial velocity, $$ u = 0 $$ (since the lemon starts from rest)

---

### **A) Time to Reach the Ground**

To find the time ($$ t $$) it takes for the lemon to reach the ground, we can use the following kinematic equation for free fall:

$$
h = \frac{1}{2} g t^2
$$

**Step 1:** Plug in the known values into the equation.

$$
3.15 = \frac{1}{2} \times 9.8 \times t^2
$$

**Step 2:** Simplify the equation to solve for $$ t^2 $$.

$$
3.15 = 4.9 \times t^2
$$
$$
t^2 = \frac{3.15}{4.9}
$$
$$
t^2 \approx 0.6429
$$

**Step 3:** Take the square root of both sides to find $$ t $$.

$$
t = \sqrt{0.6429}
$$
$$
t \approx 0.802 \text{ seconds}
$$

**Answer:**  
The lemon takes approximately **0.802 seconds** to reach the ground.

---

### **B) Speed Right Before Hitting the Ground**

To find the final speed ($$ v $$) of the lemon just before impact, we can use the following kinematic equation:

$$
v = g t
$$

Alternatively, using energy conservation principles, you can also use:

$$
v = \sqrt{2 g h}
$$

We'll use both methods to verify the result.

---

**Method 1: Using $$ v = g t $$**

**Step 1:** Plug in the known values.

$$
v = 9.8 \times 0.802
$$
$$
v \approx 7.86 \text{ m/s}
$$

---

**Method 2: Using $$ v = \sqrt{2 g h} $$**

**Step 1:** Plug in the known values.

$$
v = \sqrt{2 \times 9.8 \times 3.15}
$$
$$
v = \sqrt{61.74}
$$
$$
v \approx 7.86 \text{ m/s}
$$

**Answer:**  
The lemon's speed right before it hits the ground is approximately **7.86 meters per second (m/s)**.

---

### **Summary**

- **A)** Time to reach the ground: **0.802 seconds**
- **B)** Speed before impact: **7.86 m/s**

**Note:**  
These calculations assume that air resistance is negligible and that the acceleration due to gravity is constant at $$ 9.8 \, m/s² $$.
**A) Time to Reach the Ground:** The lemon takes approximately **0.802 seconds** to reach the ground. **B) Speed Before Impact:** The lemon's speed right before hitting the ground is approximately **7.86 meters per second (m/s)**.

Example: Lemon drop A lemon drops from a tree and falls to the ground 3.15 m below. A) How much time does it take for the lemon to reach the ground? B) What is the lemon's speed right before it hits the ground?

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