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) = limdt->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
Sinst = limt->T (dx/dt)
= limt->8 d[x(t)]/dt
= limt->8 d[5t3 – 16t + 100] / dt
= limt->8 [15t2 – 16]
= 15(8)2 – 16
= 15(64) – 16
Sinst = 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 = limt->T (dx/dt)
= limt->2.5 d[x(t)]/dt
= limt->2.5 d[4.9t2] / dt
= limt->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
Sinst = limt->T (dx/dt)
= limt->2 d[x(t)]/dt
= limt->2 d[2t2 + t + 10] / dt
= limt->2 [4t + 1]
= 4(2) + 1
Sinst = 9 cm/s
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