Which of the following equation of motion can be used to determine distance or displacement travelled by a body directly?
Both v 2 - u 2 = 2as and s = ut + 1 / 2 at 2
Understanding Equations of Motion for Distance and Displacement
Equations of motion are mathematical formulas that describe the relationship between the displacement, velocity, acceleration, and time of a body in motion. When dealing with constant acceleration, we commonly use three primary equations of motion. The question asks which of these equations can directly help us find the distance or displacement traveled by a body.
Key Equations of Motion Under Uniform Acceleration
For a body moving with constant acceleration \(a\), initial velocity \(u\), final velocity \(v\), displacement \(s\), and time \(t\), the three main equations are:
- First Equation: \(v = u + at\)
- Second Equation: \(s = ut + \frac{1}{2}at^2\)
- Third Equation: \(v^2 - u^2 = 2as\)
Analyzing Equations for Distance/Displacement
We need to identify which of these equations explicitly includes the term representing distance or displacement, which is \(s\).
- Equation 1: \(v = u + at\)
This equation relates final velocity (\(v\)), initial velocity (\(u\)), acceleration (\(a\)), and time (\(t\)). It does not contain the displacement term (\(s\)). Therefore, it cannot be used directly to determine distance or displacement. - Equation 2: \(s = ut + \frac{1}{2}at^2\)
This equation directly gives the displacement (\(s\)) in terms of initial velocity (\(u\)), time (\(t\)), and acceleration (\(a\)). If \(u\), \(a\), and \(t\) are known, \(s\) can be calculated directly. - Equation 3: \(v^2 - u^2 = 2as\)
This equation relates final velocity (\(v\)), initial velocity (\(u\)), acceleration (\(a\)), and displacement (\(s\)). If \(v\), \(u\), and \(a\) are known, \(s\) can be calculated directly using the formula \(s = \frac{v^2 - u^2}{2a}\).
Based on this analysis, both the second and third equations of motion include the displacement term (\(s\)) and can be used directly to calculate the distance or displacement traveled by a body under uniform acceleration.
Evaluating the Given Options
Let's look at the provided options in the context of our analysis:
- Option 1: \(v^2 - u^2 = 2as\)
This is the third equation of motion, which we identified as directly involving \(s\). So, this equation can be used. - Option 2: \(s = ut + \frac{1}{2}at^2\)
This is the second equation of motion, which we identified as directly involving \(s\). So, this equation can be used. - Option 3: \(v = u + at\)
This is the first equation of motion. It does not involve \(s\), so it cannot be used directly to find distance or displacement. - Option 4: Both \(v^2 - u^2 = 2as\) and \(s = ut + \frac{1}{2}at^2\)
This option correctly identifies both the second and third equations of motion as capable of directly determining distance or displacement.
Therefore, the equations that can be used directly to determine distance or displacement traveled by a body are \(s = ut + \frac{1}{2}at^2\) and \(v^2 - u^2 = 2as\).
Revision Table: Equations of Motion
| Equation | Formula | Variables Involved | Directly finds s? |
|---|---|---|---|
| First Equation | \(v = u + at\) | \(v\), \(u\), \(a\), \(t\) | No |
| Second Equation | \(s = ut + \frac{1}{2}at^2\) | \(s\), \(u\), \(a\), \(t\) | Yes |
| Third Equation | \(v^2 - u^2 = 2as\) | \(v\), \(u\), \(a\), \(s\) | Yes |
Additional Information: Displacement vs. Distance
It's important to remember the difference between distance and displacement. Displacement (\(s\)) is a vector quantity representing the shortest path from the initial to the final position. Distance is a scalar quantity representing the total length of the path traveled. The equations of motion, strictly speaking, calculate displacement. However, for motion in a single direction without changing direction, the magnitude of displacement is equal to the distance traveled.
These equations are fundamental tools in kinematics, allowing us to predict and analyze the motion of objects under constant acceleration, such as falling objects or vehicles accelerating uniformly.
Important Questions from Kinematic equations for uniformly accelerated motion
Which of the following best describes the relationship between distance, time, and speed when a body is NOT accelerating?
If the distance travelled by a body in the $n^{th}$ second is given by $(7 + 5n)$ m, then find the initial velocity and acceleration of the body respectively.
- A particle is released from height $S$ from the surface of the Earth. At a certain height, its speed is half the speed it would have just before hitting the ground. The height from the surface of Earth and the ratio of its kinetic energy to its potential energy at that instant are respectively:
- A ball is thrown vertically upward from the top of a tower. It passes the point of projection (top of the tower) moving downwards with a speed of $20 \text{ m/s}$, and eventually hits the ground with a speed of $80 \text{ m/s}$. The height of the tower is: (Take $g = 10 \text{ m/s}^2$)
- A body starts from rest with a uniform acceleration. If it travels distance $s_1$ in the first 2 seconds and distance $s_2$ in the next 4 seconds, then the relation between $s_1$ and $s_2$ will be:
