## Difference Between Speed And Velocity

- The ‘distance’ travelled by a body in unit time interval is called its
**speed**. When the position of a body changes in particular direction, then speed is denoted by ‘velocity’. i.e. the rate of change of displacement of a body is called its**Velocity**. - Speed is a scalar quantity while velocity is a vector quantity.
- ( text{Speed}=frac{text{distance}}{text{time}} )
- ( text{Velocity}=frac{text{displacement}}{text{time}} )
- Unit: In M.K.S. system = ms
^{-1}

In C.G.S. system = cm/s - If time distance graph is given then speed can be given by the slope of the line, at given time

( text{V}=frac{text{ }!!Delta!!text{ s}}{text{ }!!Delta!!text{ t}}=text{Slope} )

- The area of velocity time graph gives displacement travelled.

**Types of speed**

**(a) Average and Instantaneous speed**

**Average speed: **It is obtained by dividing the total distance travelled by the total time interval. i.e.

( text{Average speed}=frac{text{total},,text{distance}}{text{total},,text{time}} )

( text{Average},text{velocity}=frac{text{displacement}}{text{total},,text{time}} )

- Average speed is a scalar, while average velocity is a vector.
- For a moving body average speed can never be –ve or zero (unless t → ∞), while average velocity can be i.e

( >0text{ while }overset{to }{mathop}},>=or<text{ }0 ) - In general average speed is not equal to magnitude of average velocity. However it can be so if the motion is along a straight line without change in direction
- If a particle travels distances L
_{1}, L_{2}, L_{3}at speeds v_{1}, v_{2}, v_{3}etc respectively, then

( =frac{Delta s}{Delta t}=frac++…..+}{frac}}+frac}}+….+frac}}}=frac{sum{Li}}{sum{frac}}}} ) - If a particle travels at speeds v
_{1}, v_{2}etc for intervals t_{1}, t_{2}etc respectively, then

( =frac++….}++….}=frac{sum}}{sum}} )

**Instantaneous speed: **The speed of a body at a particular instant of time is called its instantaneous speed.

( =&#xnder;set{Delta tto 0}{mathop{lim }},,frac{Delta s}{Delta t}=frac{ds}{dt} )

**(b) Uniform and Non uniform speed**

**Uniform speed: **If an object covers equal distance in equal interval of time, then time speed graph of an object is a straight line parallel to time axis then body is moving with a uniform speed.

**Non-uniform speed: **If the speed of a body is changing with respect to time it is moving with a non-uniform speed.

## Speed And Velocity Example Problems With Solutions

**Example 1. **The distance between two points A and B is 100 m. A person moves from A to B with a speed of 20 m/s and from B to A with a speed of 25 m/s. Calculate average speed and average velocity.

**Solution: (i)** Distance from A to B = 100 m

Distance from B to A = 100 m

Thus, total distance = 200 m

Time taken to move from A to B, is given by

( =frac{text{distance}}{text{velocity}}=frac{100}{20}=5text{ seconds} )

Time taken from B to A, is given by

( =frac{text{distance}}{text{velocity}}=frac{100}{25}=4text{ seconds} )

Total time taken = t_{1} + t_{2} = 5 + 4 = 9 sec.

∴ Average speed of the person

( =frac{text{Total},text{dis},text{tan},text{cecovered}}{text{Total},text{time},text{taken}}=frac{200}{9}=22.2text{ m/s} )

**(ii)** Since person comes back to initial position A, displacement will be zero, resulting zero average velocity.

**Example 2. **A car moves with a speed of 40 km/hr for first hour, then with a speed of 60 km/hr for next (1frac{1}{2}) half hour and finally with a speed of 30 km/hr for next hours. Calculate the average speed of the car.

**Solution: **Distance travelled in first hour, is given by

s_{1} = speed × time = 40 km/hr × 1 hr = 40 km

Distance travelled in next half an hour, is given by

s_{2} = speed × time = 60 km/hr × (frac { 1 }{ 2 }) hr = 30 km

Distance travelled in last (1frac{1}{2}) hours, is given by

s_{3} = speed × time = 30 km/hr × (frac { 3 }{ 2 }) hr = 45 km

Thus, total distance travelled = s_{1} + s_{2} + s_{3}

= 40 + 30 + 45 = 115 km

Total time taken = 1 + (frac { 1 }{ 2 }) + (1frac{1}{2}) = 3 hours

Average speed = (frac { Total distance covered }{ Total time taken } ) = (frac { 115km }{ 3hrs })

= 38.33 km/hr

**Example 3. **Figure shows time distance graph of an object. Calculate the following :

(i) Which part of the graph shows that the body is at rest ?

(ii) Average speed in first 10 s.

(iii) Speeds in different parts of motion.

**Solution: (i)** The part BC shows that the body is at rest.

**(ii)** In first 10 seconds, distance travelled = 100m

( text{Average speed}=frac{text{total},,text{distance}}{text{total},,text{time}} )

( =frac{100}{10}=10text{ m/s} )

**(iii)** Speed of the object in part AB is given by slope = 100/6 = (frac { 50 }{ 3 }) m/s

Speed of object in part BC = 0 m/s

Speed of the object in part CD

( =frac{100-40}{12-10}=frac{60}{2}=30~text{m/s} )

Speed of object in part DE

( =frac{40-0}{14-12}=frac{40}{2}=30~text{m/s} )

**Example 4. **Time-velocity graph of a particle is shown in Figure. Calculate the distance travelled in first seconds.

**Solution: **Distance travelled in first 8s is given by area OABCG

= area of rectangle OAMG + area of triangle BMC

= 8 × 60 + (frac { 1 }{ 2 }) × 4 × 40

= 480 + 80 = 560 m.

**Example 5. **A cow walked along a curved path from P to Q, which is 70 m away from P. Q lies to the south-west of P. The distance travelled by the cow is 240 m and the time taken is 160 s.

Calculate the

(a) average speed,

(b) average velocity,

of the cow moving from P to Q.

**Solution:**

Total distance travelled = 240 m

Displacement = 70 m

Time taken = 160 s

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