Motion in a Straight Line
IMAT Interactive Study Tool
Distance vs. Displacement
Distance is the total path length an object travels. It's a scalar quantity, meaning it only has magnitude (e.g., 10 meters).
Displacement is the object's overall change in position, from its start point to its end point. It's a vector quantity, meaning it has both magnitude and direction (e.g., 2 meters East).
Speed vs. Velocity
Speed is the rate of change of distance. It is a scalar.
Velocity is the rate of change of displacement. It is a vector.
Average Speed = \(\frac{\text{Total Distance}}{\text{Total Time}}\)
Average Velocity = \(\frac{\text{Total Displacement}}{\text{Total Time}} = \frac{\Delta x}{\Delta t}\)
Acceleration
Acceleration is the rate of change of velocity. It is a vector. An object accelerates if its speed, direction, or both change.
Average Acceleration = \(\frac{\text{Change in Velocity}}{\text{Total Time}} = \frac{\Delta v}{\Delta t}\)
1. True or False: An object can have a constant speed but a changing velocity.
2. True or False: If an object's acceleration is zero, it must be stationary.
Example 1: Constant Acceleration
A train starts from rest and accelerates uniformly at \(2 \, m/s^2\) for 10 seconds. How far does it travel in this time?
Step 1: Identify the knowns and the unknown.
We are given: Initial velocity (\(u\)) = 0 (starts from rest), Acceleration (\(a\)) = \(2 \, m/s^2\), Time (\(t\)) = 10 s. We need to find the displacement (\(s\)).
Step 2: Choose the appropriate kinematic equation.
The equation that relates \(s, u, a,\) and \(t\) is \(s = ut + \frac{1}{2}at^2\).
Step 3: Substitute the values and solve.
\(s = (0)(10) + \frac{1}{2}(2)(10)^2 = 0 + \frac{1}{2}(2)(100) = 100 \, m\)
Conclusion: The train travels 100 meters.
Example 2: Average Speed vs. Average Velocity
A runner jogs 400 m East and then 300 m North. The entire jog takes 100 seconds. Find the runner's average speed and average velocity.
Step 1: Calculate the total distance and average speed.
Total Distance = 400 m + 300 m = 700 m.
Average Speed = \(\frac{700 \, m}{100 \, s} = 7 \, m/s\)
Step 2: Calculate the total displacement.
The displacement is the hypotenuse of a right-angled triangle with sides 400 m and 300 m. We use Pythagoras' theorem.
Displacement = \(\sqrt{400^2 + 300^2} = \sqrt{160000 + 90000} = \sqrt{250000} = 500 \, m\).
Step 3: Calculate the average velocity.
Average Velocity = \(\frac{500 \, m}{100 \, s} = 5 \, m/s\)
Conclusion: The average speed is 7 m/s, while the magnitude of the average velocity is 5 m/s.
| Quantity | Type | Definition | Unit (SI) |
|---|---|---|---|
| Distance | Scalar | Total path length | meter (m) |
| Displacement | Vector | Change in position | meter (m) |
| Speed | Scalar | Rate of change of distance | m/s |
| Velocity | Vector | Rate of change of displacement | m/s |
| Acceleration | Vector | Rate of change of velocity | m/s² |