a lemming springs vertically upwards with a velocity of 1.4m/s off the top of a 50m high cliff, how long does it take the lemming to hit the bottom

We can solve this problem by breaking it into two parts using the principles of kinematics.

### Variables:
- Initial velocity (\( u \)) = \( 1.4 \, \text{m/s} \) (upwards)
- Height of the cliff (\( h \)) = \( 50 \, \text{m} \)
- Gravitational acceleration (\( g \)) = \( 9.8 \, \text{m/s}^2 \) (downwards)
- Total time taken (\( t \)) = ?

### Approach:
1. **First phase**: The lemming moves upwards until it reaches its highest point.
2. **Second phase**: It falls down from the highest point back to the ground.

#### 1. Time to reach the highest point:
At the highest point, the lemming's velocity becomes zero. Using the equation:
\[
v = u - g t_1
\]
where \( v = 0 \, \text{m/s} \), we can solve for \( t_1 \), the time taken to reach the highest point:
\[
0 = 1.4 - 9.8 t_1
\]
\[
t_1 = \frac{1.4}{9.8} \approx 0.143 \, \text{seconds}
\]

#### 2. Maximum height reached:
Now, using the kinematic equation:
\[
h_1 = u t_1 - \frac{1}{2} g t_1^2
\]
\[
h_1 = 1.4(0.143) - \frac{1}{2}(9.8)(0.143)^2
\]
Let's calculate \( h_1 \), and then add it to the cliff height to find the total distance the lemming falls down.The lemming reaches a maximum height of approximately \( 50.1 \, \text{m} \) from the bottom of the cliff (including the height of the cliff). The time taken to reach this maximum height is \( 0.143 \, \text{seconds} \).

#### 3. Time to fall from the maximum height:
Now, we calculate the time it takes to fall from the maximum height of \( 50.1 \, \text{m} \). We can use the equation for free fall:
\[
h = \frac{1}{2} g t_2^2
\]
where \( h = 50.1 \, \text{m} \) and \( g = 9.8 \, \text{m/s}^2 \). Solving for \( t_2 \), the time to fall, we get:
\[
t_2 = \sqrt{\frac{2h}{g}}
\]

Let's calculate \( t_2 \).The time taken to fall from the maximum height of 50.1 meters is approximately \( 3.20 \, \text{seconds} \).

#### 4. Total time:
The total time for the lemming to hit the ground is the sum of the time taken to reach the highest point and the time to fall back down:
\[
t_{\text{total}} = t_1 + t_2 = 0.143 + 3.20 \approx 3.34 \, \text{seconds}
\]

Thus, it takes the lemming approximately **3.34 seconds** to hit the bottom of the cliff.

Would you like more details or explanations? Here are some related questions to explore:

1. How does increasing the initial velocity affect the total time?
2. What would happen if the cliff were higher, say 100 meters?
3. How would air resistance change the calculation?
4. What is the maximum height the lemming reaches before falling?
5. How would the time change if gravity on another planet were different?

**Tip:** Always keep in mind the direction of motion when assigning signs to velocity and acceleration in kinematic equations.

A lemming springs vertically upwards with a velocity of 1.4m/s off the top of a 50m high cliff, how long does it take the lemming to hit the bottom?

Discover how to calculate the time it takes for a lemming to hit the bottom after jumping upwards with a velocity of 1.4 m/s from a 50m cliff. Learn step-by-step how to apply kinematic equations and free-fall motion to solve this physics problem. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation