which equation would you use to find the distance an object has travelled?
LectureNotes said which equation would you use to find the distance an object has travelled?
Answer:
To determine the distance an object has traveled, you can use different equations depending on the context of the motion and the information available. Here are some common scenarios and their corresponding equations:
1. Constant Speed:
If the object is traveling at a constant speed, the distance (d) can be calculated using the equation:
d = v \times t
where:
- d is the distance traveled,
- v is the constant speed (velocity),
- t is the time of travel.
2. Uniformly Accelerated Motion:
If the object is accelerating uniformly, you can use the following kinematic equations. One common equation is:
d = v_i \times t + \frac{1}{2} a \times t^2
where:
- d is the distance traveled,
- v_i is the initial velocity,
- a is the acceleration,
- t is the time of travel.
3. Free Fall (Under Gravity):
For an object in free fall (ignoring air resistance), the distance (d) fallen can be calculated using:
[ d = \frac{1}{2} g \times t^2 ]
where:
- d is the distance fallen,
- g is the acceleration due to gravity (approximately 9.8 \, \text{m/s}^2 on Earth),
- t is the time of fall.
4. Displacement with Initial and Final Velocity:
If you know the initial velocity (v_i), final velocity (v_f), and the time (t), you can use the average velocity to find the distance:
d = \frac{(v_i + v_f)}{2} \times t
where:
- d is the distance traveled,
- v_i is the initial velocity,
- v_f is the final velocity,
- t is the time of travel.
5. Using Velocity and Acceleration:
If you know the initial velocity (v_i), acceleration (a), and the distance (d), you can use the following equation to find the final velocity (v_f), which can then help in calculating the distance:
v_f^2 = v_i^2 + 2a \times d
Rearranging this equation to solve for d gives:
d = \frac{v_f^2 - v_i^2}{2a}
Example Problem:
Let’s solve an example using the uniformly accelerated motion equation. Suppose an object starts from rest (v_i = 0), accelerates at 3 \, \text{m/s}^2 for 5 seconds. The distance traveled can be calculated as follows:
d = v_i \times t + \frac{1}{2} a \times t^2
d = 0 \times 5 + \frac{1}{2} \times 3 \times 5^2
d = 0 + \frac{1}{2} \times 3 \times 25
d = \frac{3 \times 25}{2}
d = \frac{75}{2}
d = 37.5 \, \text{meters}
Therefore, the object travels 37.5 meters.
By understanding the context and the given variables, you can choose the appropriate equation to find the distance an object has traveled.