ANAND CLASSES study material and notes boost your JEE, NEET, and CBSE Class 11 exam preparation with these MCQs on Average Velocity. Each question includes step-by-step solutions and explanations to help you master this fundamental physics concept.
Average Velocity
Velocity is a vector quantity that defines the rate at which an object changes its position. Unlike speed, velocity includes direction. If an object covers different displacements with varying speeds or takes different time intervals, we calculate the average velocity over the entire journey.
Average velocity helps in understanding the overall motion of an object when the velocity is not constant.
Definition :
The average velocity of a body for a given interval of time is defined as the ratio of the total displacement traveled to the total time taken.
Mathematically,
$$v_{av} = \frac{\text{Total Displacement Travelled}}{\text{Total Time Taken}}$$
Understanding Average Velocity with an Example
Imagine a car travels 180 km in 3 hours along North and then another 120 km in 2 hours along South. The total displacement traveled is: 180 – 120 = 60 km
The total time taken is: 3 + 2 = 5 hours
Thus, the average velocity is: $$v_{av} = \frac{60}{5} = 12 \text{ km/h}$$
This means the car covered an average of 12 km per hour along North direction over the entire journey.
Types of Average Velocity
1. Time-Averaged Velocity
When a particle moves at different uniform velocities $v_1, v_2, v_3, \dots$ for different time intervals $t_1, t_2, t_3, \dots$, then:
$$v_{av} = \frac{\text{Total Displacement Travelled}}{\text{Total Time Taken}}$$
$$v_{av} = \frac{d_1 + d_2 + d_3 + \dots}{t_1 + t_2 + t_3 + \dots}$$
$$v_{av} = \frac{v_1t_1 + v_2t_2 + v_3t_3 + \dots}{t_1 + t_2 + t_3 + \dots}$$
Note : Time average velocity is the arithmetic mean of different velocities over a given time period.
It is used when an object moves with different velocities for equal time intervals.
Example: A bus moves at 80 km/h for 2 hours in east direction, then at 40 km/h for 3 hours in west direction. What is the average velocity of bus ?
Solution :
The total displacement traveled: (80×2)-(40×3)=160-120=40 km
Total time taken: 2+3=5 hours
So, the average velocity: $$v_{av} = \frac{40}{5} = 8 \text{ km/h}$$
This means the bus covered an average of 8 km per hour along East direction over the entire journey.
Special Case:
If a particle moves with velocity $v_1$ for half the total time and $v_2$ for the remaining half, then: $$v_{av} = \frac{v_1 + v_2}{2}$$
2. Displacement-Averaged Velocity
If a particle covers different displacements $d_1, d_2, d_3, \dots$ with velocities $v_1, v_2, v_3, \dots$, then:
$$v_{av} = \frac{\text{Total Displacement Travelled}}{\text{Total Time Taken}}$$
$$v_{av} = \frac{d_1 + d_2 + d_3 + \dots}{t_1 + t_2 + t_3 + \dots}$$
$$v_{av} = \frac{d_1 + d_2 + d_3 + \dots}{\frac{d_1}{v_1} + \frac{d_2}{v_2} + \frac{d_3}{v_3} + \dots}$$
Note : Displacement average velocity (also known as harmonic mean velocity) is calculated when different displacements are covered with different velocities.
It is useful in cases where an object covers equal distances at different velocities.
Special Cases:
- If a particle moves half the total displacement at $v_1$ and the other half at $v_2$, then:
$$v_{av} = \frac{2 v_1 v_2}{v_1 + v_2} $$
- If a particle moves one-third of the displacement at $v_1$, the next one-third at $v_2$, and the last one-third at $v_3$, then:
$$v_{av} = \frac{3 v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1} $$
Difference Between Average Speed and Average Velocity
| Feature | Average Speed | Average Velocity |
|---|---|---|
| Definition | Total distance traveled divided by total time taken. | Total displacement divided by total time taken. |
| Formula | $v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}}$ | $v_{\text{avg}} = \frac{\text{Total Displacement}}{\text{Total Time}}$ |
| Type of Quantity | Scalar (only magnitude, no direction). | Vector (has both magnitude and direction). |
| Depends On | Total path length covered. | Straight-line distance between initial and final position. |
| Can be Zero? | Never zero if there is motion. | Can be zero if initial and final positions are the same. |
| Always Positive? | Yes, it is always positive. | Can be positive, negative, or zero, depending on displacement. |
| Example | A car moves 60 km forward and 40 km back in 2 hours. Total distance = 100 km, Total time = 2 hr. Average speed = $\frac{100}{2}$ = 50 km/h. | The same car’s displacement = 20 km (net position change). Average velocity = $\frac{20}{2}$ = 10 km/h in the forward direction. |
📌 Key Takeaways:
- Average speed measures how fast an object moves, regardless of direction.
- Average velocity considers only the net position change and can be zero if the object returns to its starting point.
FAQs
Q1: Is average velocity a vector or a scalar?
A: Average velocity is a vector quantity because it depends on both the magnitude of the total displacement traveled, and on direction.
Q2: Why is average speed always greater than or equal to average velocity?
A: Since displacement is always ≤ distance, and velocity is based on displacement, the magnitude of average velocity is always ≤ average speed.
Q3: Is average velocity always equal to the arithmetic mean of velocities?
A: No, it depends on whether time intervals are equal (arithmetic mean) or distances are equal (harmonic mean).
Q4: Why is distance average velocity always less than time average velocity?
A: Because harmonic mean is always less than or equal to the arithmetic mean.
Multiple Choice Questions (MCQs) with Answers and Explanations
Q1: A car travels 100 m in 5 seconds and then 150 m in 10 seconds. What is its average velocity?
(a) 10 m/s
(b) 12 m/s
(c) 15 m/s
(d) 17 m/s
Answer: (b) 12 m/s
Explanation: $$v_{av} = \frac{\text{Total Distance}}{\text{Total Time}}$$
$$v_{av} = \frac{100 + 150}{5 + 10} = \frac{250}{15} = 12 \text{ m/s}$$
Q2: A bus moves at 40 km/h for 2 hours and then at 60 km/h for 3 hours. What is its average velocity?
(a) 48 km/h
(b) 50 km/h
(c) 52 km/h
(d) 54 km/h
Answer: (c) 52 km/h
Explanation:
Total displacement traveled: (40×2)+(60×3)=80+180=260 km
Total time taken: 2+3=5 hours
Average velocity
$$v_{av} = \frac{\text{Total Displacement}}{\text{Total Time}}$$
$$v_{av} = \frac{260}{5} = 52 \text{ km/h}$$
Q3: A train moves half the total displacement at 30 km/h and the other half at 60 km/h. What is its average velocity?
(a) 40 km/h
(b) 42 km/h
(c) 45 km/h
(d) 48 km/h
Answer: (a) 40 km/h
Explanation: Using the displacement-averaged velocity formula:
$$v_{av} = \frac{2 v_1 v_2}{v_1 + v_2} $$
$$v_{av}= \frac{2 \times 30 \times 60}{30 + 60}$$
$$v_{av} = \frac{3600}{90} = 40 \text{ km/h}$$
Q4: A cyclist covers one-third of the total distance at 20 km/h, next one-third at 30 km/h, and the last one-third at 60 km/h. What is the average velocity?
(a) 30 km/h
(b) 32 km/h
(c) 35 km/h
(d) 40 km/h
Answer: (b) 32 km/h
Explanation:
Using the formula for three equal distances:
$$v_{av} = \frac{3 v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1}$$
$$v_{av} = \frac{3 \times 20 \times 30 \times 60}{(20 \times 30) + (30 \times 60) + (60 \times 20)}$$
$$v_{av} = \frac{108000}{3360} = 32.14 \approx 32 \text{ km/h}$$
Q5: A person walks 2 km at 5 km/h, then jogs 3 km at 10 km/h. What is the average velocity?
(a) 6 km/h
(b) 6.5 km/h
(c) 7 km/h
(d) 7.5 km/h
Answer: (c) 7 km/h
Explanation:
Total distance traveled: 2+3=5 km
Time taken for first part: $$t_1=\frac{2}{5} = 0.4 \text{ hours}$$
Time taken for second part: $$t_2=\frac{3}{10} = 0.3 \text{ hours}$$
Total time taken: 0.4+0.3=0.7 hours
Average velocity:
$$v_{av} = \frac{\text{Total Displacement}}{\text{Total Time}}$$
$$v_{av} = \frac{5}{0.7} = 7.14 \approx 7 \text{ km/h}$$
Q6: A car travels 100 m in 5 seconds and then 150 m in 10 seconds. What is its average speed?
(a) 10 m/s
(b) 12 m/s
(c) 15 m/s
(d) 17 m/s
Answer: (b) 12 m/s
Explanation: Average velocity:
$$v_{av} = \frac{\text{Total Displacement}}{\text{Total Time}}$$
$$v_{av} = \frac{(100 + 150)}{(5 + 10)} = \frac{250}{15} = 12 \text{ m/s}$$
These MCQs help in understanding how to calculate velocity speed under different conditions, which is crucial for JEE, NEET, and CBSE Board Class 11 exams.
Conceptual Questions with Answers
Q1: Can average velocity be zero?
A: Yes, if an object returns to its starting position, the total displacement is zero, making the average velocity zero, even if the object has traveled a significant distance.
Q2: What is the key difference between time average velocity and distance average velocity?
A1: Time average velocity considers the arithmetic mean of different velocities over equal time intervals, while distance average velocity considers the harmonic mean of different velocities over equal distances.
Do You Know?
- Average speed and velocity are the same only for straight-line motion without turning back.
- If a body moves at a constant speed in a circular path, its average velocity over one complete revolution is zero.
- If an object moves with two different velocities for equal time intervals, the average velocity is their arithmetic mean.
- If an object moves with two different velocities for equal distances, the average velocity is their harmonic mean.
Worksheet on Average Velocity
- A train travels at 60 km/h for 2 hours and then at 80 km/h for 3 hours. Find the average velocity.
- A person walks 100 m at 5 m/s and another 200 m at 10 m/s. Calculate the average velocity.
Test Paper (10 Marks)
- Define average velocity. (2 Marks)
- Differentiate between distance average velocity and time average velocity. (3 Marks)
- A car moves with speeds of 40 km/h and 60 km/h for equal distances. Find the average velocity. (5 Marks)
Quick Revision Points
✔ Average velocity is total displacement divided by total time.
✔ Time average velocity is the arithmetic mean of velocities.
✔ Distance average velocity is the harmonic mean of velocities.
✔ If total displacement is zero, the average velocity is zero.
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