Instantaneous Speed Formula

Velocity is defined as the rate of change of its position with respect to its frame of reference. It is a vector quantity as it has magnitude and direction. The SI unit of velocity is meter per second or m/s.Whereas speed measures the distance traveled by an object over the change in time. It has magnitude but no direction and thus is a scalar quantity. The SI unit of speed is a meter per second or m/s.

Introduction to Instantaneous Speed

Instantaneous speed is defined as the speed of an object at a particular instant in time. It is the magnitude of velocity. It is the actual speed at a particular moment. As the time approaches zero, the distance traveled by an object also approaches zero. It is the non-zero limit of the distance to time ratio. In terms of a graph, instantaneous speed is the slope of the tangent at any point in the journey. 

The formula for Instantaneous Speed:

As per the formula, instantaneous speed is the ratio of distance upon a time.

Speed(i) = lim_dt->0 ds/dt

where,

Speed(i) = Instantaneous speed

ds = Distance traveled

dt = Time interval

Instantaneous speed can be calculated by dividing the shortest distance covered by an object in a short time interval. we can also calculate it by determining the slope of a position versus a time graph.

Unit of Instantaneous Speed

The SI unit of instantaneous speed is meter per second or m/s. The CGS unit of instantaneous speed is cm/s. It is a scalar quantity because it has magnitude but no direction. 

Difference between Average speed and Instantaneous speed

Average Speed	Instantaneous Speed
Average speed is defined as the total distance travelled divided by the total elapsed time.	Instantaneous speed is defined as the speed at a particular instant of time.
Formula: Total distance / Total time	Formula: The distance at that instant / Time at that instant.
It is measured by calculating the speed for an entire journey	It is measured with the help of a speedometer.
It is always constant.	It is not constant.
Example: A bike travelling with a speed of 40 miles/hour; thus, the average speed is 40 miles in an hour	Example: A bike travelling at a certain speed at an instant of time can be given by a speedometer.

Difference between Instantaneous Speed and Instantaneous Velocity

Instantaneous Speed	Instantaneous Velocity
Instantaneous speed is the magnitude of instant velocity at a given instant of time	Instantaneous velocity is the change of position that takes place at a very small interval of time
It is a scalar quantity.	It is a vector quantity.
Formula: Speed(I) = ds/dt	Formula: Vi = lim∆t→0 ds/dt
Unit: Meters per second (m/s)	Unit: Meters per second (m/s)

Sample Problems

Question 1: Calculate the instantaneous speed for an object that travels the distance given by the function x(t) = 5t³ – 16t +100 m at t=8s.

Solution:

Given: 

x(t) = 5t³ – 16t +100 m 

t = 8s

S_inst = lim_t->T (dx/dt)

=  lim_t->8 d[x(t)]/dt

= lim_t->8 d[5t³ – 16t + 100] / dt

= lim_t->8 [15t² – 16]

= 15(8)² – 16

= 15(64) – 16

S_inst = 944 m/s

Question 2: A telescope takes a picture of a meteor traveling a distance of 100 km in 0.001 seconds. What is the instantaneous speed of this meteor at the instant the picture is taken? 

Solution:

In the very short time duration, the instantaneous speed of the meteor will be given by:

Speed = Distance/Time 

= 100 km/0.001 seconds 

= 1,00,000 km/seconds. 

Question 3:  A ball is thrown up in the air. It goes all the way up, and then at time t = a units, it stops traveling upwards and starts its journey back down. What will be the instantaneous speed of the ball at the time t = a units? 

Answer:

At the instant of t = a units, the instantaneous speed of the ball will be zero as the ball stops and then starts its journey downwards under the force of gravity.

Question 4: When an object is dropped and acted on by gravity, its position changes according to the function x(t) = 4.9t², and x(t) is in units of meters. What is the instantaneous speed at t = 2.5 s?

Solution:

Given:

x(t) = 4.9t²

t = 2.5 s

Find the instantaneous speed by using the formula:

Sinst = lim_t->T (dx/dt)

=  lim_t->2.5 d[x(t)]/dt

= lim_t->2.5 d[4.9t²] / dt

= lim_t->2.5 [9.8t]

= 9.8(2.5)

= 24.5 m/s

Question 5:  Calculate the instantaneous speed for an object that travels the distance given by the function x(t) = 2t² + t + 10 cm at t = 2s.

Answer:

Given:

x(t) = 2t² + t + 10 cm

t = 2s

S_inst = lim_t->T (dx/dt)

= lim_t->2 d[x(t)]/dt

= lim_t->2 d[2t² + t + 10] / dt

= lim_t->2 [4t + 1]

= 4(2) + 1

S_inst = 9 cm/s