Initial Velocity Formula

Last Updated : 4 Feb, 2024

The first thing that comes to mind after hearing initial velocity is that it is the velocity of an object when it started moving from rest. normally we do this mistake, but in reality, it is the velocity of an object at a given instance of time (that can be assumed) which it got before being under the effect of acceleration after some instance of time (which can be also be assumed). Let's learn why? and how?.

First equation of motion states that ,

v = u + at

Where v is final velocity, u is initial velocity, t is time(time instance) and a is acceleration.

Now let's rearrange this equation,

u = v - at,

Here v will have some value it can be zero, positive or negative. so, what does it prove? it's not necessary that an object needs to be at rest for a velocity to be initial velocity 'u'.
If a = 0, acceleration is zero. now what? initial velocity is equal to final velocity. so at any value of 't' will not make any difference. so that defines that initial velocity is the velocity of an object before being under the effect of acceleration 'a'.
If we assume at time instance t= 0, then the same thing happens. initial velocity is equal to final velocity. and here we decided to keep time instant t = 0. but for a fact, we know time is never zero so we just consider a time instant where we can assume time t = 0.

Now we are having a good base on initial velocity, we can finally move to the "Initial velocity formula".

Initial velocity formula

Basically while finding the initial velocity of an object, not all the elements needed to find it is not present. so, we need different formulas to find the initial velocity within the limit of elements given to us. so there are 4 possible combinations of elements that can be solved with the initial velocity formulas and we got 4 different formulas to solve them. all of these formulas are derived from the "3 equations of motion".

Three Equations of Motion are:

v = u + at ⇢ (1.1)
s = ut + ½ × at² ⇢ (1.2)
v²- u²= 2as ⇢ (1.3)

Where u is initial velocity, v is final velocity, t is time (time instance), s is distance, a is acceleration

Four Possible cases that can be given to find Initial Velocity

Let's suppose, If time (t), acceleration (a), and final velocity (v) are provided to us, for this case let's consider equation (1.1), rearranging this equation to get,
u = v - at

And now put values in this equation to find the initial velocity.

Let's suppose, If the distance (s), acceleration (a) and time(t) are provided to us, for this case let us consider equation (1.2), dividing this equation with (t) to get,
s/t = u + ½at

After rearranging this equation,

u = s/t - at/2

And now put values in this equation to find the initial velocity.

Let's suppose, velocity (v), acceleration (a), and distance(s) are provided to us, for this case let's consider equation (1.3), rearrange this equation to get,
u²= v²- 2as

And now put values in this equation to find the initial velocity.

Let's suppose, If final velocity (v), distance (s), and time (t) are provided to us, for this case let's consider equations (1.2) and (1.1). Firstly from equation 1.1 let's find the value of 'a' in known terms of 'v', 'u' and 't', by rearranging,
a = (v - u)/t,

Lets put this value of 'a' in equation 1.2,

s = ut + 1/2 × ((v - u)/t) × t²

After solving and rearranging we get,

u = 2s/t - v

And now put values in this equation to find the initial velocity.

Sample Problems

Question 1: A car is moving through traffic at a slow speed. Once traffic got cleared, the car accelerates at 0.20 ms⁻² for 60.0 s. After this acceleration, the velocity of the car is 30.0 ms⁻¹. Determine the initial velocity of the car.

Solution:

Given: t = 60.0 s, a = 0.20 ms⁻², v = 30.0 ms⁻¹

Thus, the initial velocity is:

u = v – at

Inserting the values in the formula,

u = 30 – (0.20)×(60.0)

⇒ u = 30 – 12

⇒ u = 18 ms⁻¹

⇒ u = 18.0 ms⁻¹.

Question 2: A Ship covers 1000 m in 10 seconds and has an acceleration of 10 ms^-2.find the initial velocity of the Ship.

Solution:

Given: s =1000m , t = 10s, a = 10ms^-2

Thus, the initial velocity is:

u = s/t – at/2

Inserting the values in the formula

u = 1000/10 – (10 × 10)/2

⇒ u = 100 – 50

⇒ u = 50 ms^-1.

Question 3: An athlete covers a distance of 100 m. If his final velocity was 40 ms⁻¹ and has an acceleration of 6 ms⁻². Compute her initial velocity?

Solution:

Given: Distance, s = 100m,Final velocity, v = 40 ms⁻¹, Acceleration, a = 6 ms⁻²

Thus, the initial velocity is:

u² = v² – 2as

Inserting the values in the formula

u² = 40² – 2 × 6 × 100

⇒ u² = 1600 – 1200

⇒ u²= 400

⇒ u = 20 ms⁻¹

Question 4: a person rides a bicycle at speed of 30 ms^-1and covers 500 m in 20 seconds. what was the initial velocity of the bicycle?

Solution:

Given: v = 30ms^-1, s = 500m, t = 20s

Thus, the initial velocity is,

u = 2s/t – v

Inserting the values in the formula,

u = (2 × 500)/20 – 30

⇒ u = 1000/20 – 30

⇒ u = 50 - 30

⇒ u = 20ms^-1

Question 5: In a Boat race, the winner Finishes the race at speed of 70 ms^-1 in 40sec. if Race is 1 km long. Compute the initial velocity of the boat?

Solution:

Given: v = 70ms^-1, s = 1km = 1000m, t = 40s

Thus, the initial velocity is,

u = 2s/t – v

Inserting the values in the formula,

u = (2 × 1000)/40 – 70

⇒ u = 2000/20 – 70

⇒ u = 100 - 70

⇒ u = 30ms^-1

Question 6: A motorbike covers 1.5km in 12 seconds and has an acceleration of 20 ms^-2. Find the initial velocity of the motorbike.

Solution:

Given: s = 1.5km = 1.5 × 1000 = 1500m, t = 12s, a = 20ms^-2

Thus, the initial velocity is:

u = s/t – at/2

Inserting the values in the formula

u = 1500/10 – (20 × 12)/2

⇒ u = 150 – 120

⇒ u = 30 ms^-1.

Question 7: A Bus going through a hilly area moves slowly because of sharp turns. when the bus moves to a plainer area, the driver accelerates the bus at 0.40 ms−2 for 30.0 s. After this acceleration, the velocity of the bus is 50.0 ms⁻¹. Determine the initial velocity of the bus.

Solution:

Given: t = 30.0 s, a = 0.20 ms⁻²,v = 50.0 ms⁻¹

Thus, the initial velocity is:

u = v – at

Inserting the values in the formula,

u = 50 – (0.40) × (30.0)

⇒ u = 50 – 12

⇒ u = 38 ms⁻¹

⇒ u = 38.0 ms⁻¹.