Motion IV
Speed

Speed can be defined as the rate of distance covered. It is simply the time taken to cover a certain distance. The S.I unit of speed is m/s. Speed is a scaler quantity as it doesn't give us information about direction.

$${speed} = \frac{distance}{time} $$

Instantaneous speed

Instantaneous speed can be defined as the distance covered at any instant in time.

$$ {speed_i} = \frac{ds}{dt} $$

Uniform speed

Uniform speed can be defined as the distance covered in equal time intervals. A body is said to move at a uniform speed if it covers equal distance at equal time intervals.

Distance-time graph

A distance-time graph is a graphical representation of the distance covered by a body per unit time. The path for a uniform speed on a distance-time graph is usually a straight line.

Velocity

Velocity can be defined as the rate of displacement covered. It is the time taken to cover a distance in a specified direction. It is a vector quantity. The S.I.unit of velocity is m/s. $${velocity} = \frac{displacement}{time} $$

Uniform Velocity

A body is said to move at uniform velocity if it covers equal displacement at equal time intervals.

Instantaneous velocity

Instantaneous velocity is the displacement covered by a body at any instant in time.

Displacement-time graph

This is a graphical representation of the velocity of a body. For a uniform velocity, the graph is usually a straight line.

Calculations

Example 1: The distance xm travelled by a particle in time t seconds is described by the equation x = 10 + 12t². Find the average speed of the particle between the time interval t = 2s and t = 5s. (JAMB)

Solution

$$\text{Average speed} = \frac{dx}{dt} $$ $${where,} $$ $${dx} = \text{change in distance} $$ $${dt} = \text{change in time} $$ $$\text{By differential calculus} $$ $$\text{for t = 5s} $$ $${x} = 10 + 12(5)² $$ $${x} = 10 + 300 $$ $${x} = 310m $$
$$\text{for t = 2s} $$ $${x} = 10 + 12(2)² $$ $${x} = 10 + 48 $$ $${x} = 58m $$ $${speed} = \frac{x_2 - x_1}{t_2 - t_1} $$ $${speed} = \frac{310 - 58} {5 - 2} $$ $$ {speed} = \frac{252} {3} $$ $${speed} = 84m/s $$

Example 2: A student walks a distance of 3km in 20 minutes. Calculatate his average speed

Solution

$${speed} = \frac{distance}{time} $$ Recall the S.I.unit for speed is m/s
Convert all parameters to standard unit $${distance} = {3 × 1000} = {3000m} $$ $${time} = 20 × 60 = 1200s $$ $${speed} = \frac{3000}{1200} $$ $${speed} = 2.5m/s $$

Example 3: A boy cycles continuously through a distance of 1km in 5 minutes. Calculate his average speed.

Solution

$${speed} = \frac{distance}{time} $$ $${distance} = 1000m $$ $${time} = 5 × 60 = 300s $$ $${speed} = \frac{1000} {300} $$ $${speed} = 3.33m/s $$

Summary