What is Speed?
Definition of Speed: Speed is the rate of change of distance with time. Speed is different from velocity because it’s not in a specified direction. In this article, you will learn how to solve speed, velocity, and acceleration problems.
Additionally, you need to know that speed is a scalar quantity and we can write its symbol as S. The formula for calculating the speed of an object is:
Speed, S = Distance (d) / Time (t)
Thus, s = d/t
Note: In most cases, we also use S as a symbol for distance.
The S.I unit for speed is meter per second (m/s) or ms-1
Non-Uniform or Average Speed: This is a non-steady distance covered by an object at a particular period of time. We can also define non-uniform speed as the type of distance that an object covered at an equal interval of time.
The formula for calculating non-uniform speed is
Average speed = Total distance covered by the object / Total time taken
Actual speed: This is also known as the instantaneous speed of an object which is the distance covered by an object over a short interval of time.
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What is Velocity?
Definition of Velocity: Velocity is the rate of displacement with time. Velocity is the speed of an object in a specified direction. The unit of velocity is the same as that of speed which is meter per second (ms-1). We use V as a symbol for velocity.
Note: We often use U to indicate initial speed, and V to indicate final speed.
The formula for calculating Velocity (V) = displacement (S) / time (t)
The difference between velocity and speed is the presence of displacement and distance respectively. Because displacement is a measure of separation in a specified direction, while distance is not in a specified direction. Velocity is a vector quantity.
Uniform Velocity
Definition of uniform velocity: The rate of change of displacement is constant no matter how small the time interval may be. Also, uniform velocity is the distance covered by an object in a specified direction in an equal time interval.
The formula for uniform velocity = Total displacement / Total time taken
What is Acceleration?
Definition of Acceleration: Acceleration is the rate of change of velocity with time. Acceleration is measured in meters per second square (ms-2). The symbol for acceleration is a. Acceleration is also a vector quantity.
The formula for acceleration, a = change in velocity (v)/time taken (t)
Thus, a = v/t
We can also write acceleration as
a = change in velocity/time = ΔV/Δt = (v – u)/t
[where v = final velocity, u = initial velocity, and t = time taken]
Uniform Acceleration
In the case of uniform acceleration, the rate of change of velocity with time is constant.
Average velocity of the object = (Initial velocity + final velocity)/2
or
Average velocity = (v + u)/2
What is Retardation?
Retardation is the decreasing rate of change in velocity moved, covered, or traveled by an object.
The formula for calculating retardation is
Retardation = Change in a decrease in velocity/time taken
Equations of Motion
You can also apply the following equations of motion to calculate the speed, velocity, and acceleration of the body:
- v = u + at
- s = [ (v + u)/2 ]t
- v2 = u2 + 2as
- s = ut + (1/2)at2
Where v = final velocity, u = initial velocity, t = time, a = acceleration, and s = distance
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Solved Problems of Speed, Velocity, and Acceleration
Here are solved problems to help you understand how to calculate speed, velocity, and acceleration:
Problem 1
A train moves at a speed of 54km/h for a one-quarter minute. Find the distance travelled by train.
Solution
Data
From the question above
Speed = 54 km/h = (54 x 1000)/(60 x 60) = 54,000/3,600 = 15 m/s
Time = one quarter minute = 1/4 minute = (1/4) x 60 = 15 seconds
Since we have
speed = distance/time
After cross-multiplication, we will now have
Distance = speed x time
We can now insert our data into the above expression
Distance = 15 m/s x 15 s = 225 m
Therefore, the distance travelled by train is 225 meters.
Problem 2
A car travelled a distance of 5km in 50 seconds. Find the speed in meters per second.
Solution
Data:
Distance = 5km = 5 x 1000m = 5,000m
Time = 50 seconds
speed = ?
and the formula for speed
speed = distance/time = 5000/50 = 100m/s
Problem 3
A motorcycle starting from rest moves with a uniform acceleration until it attains a speed of 108 kilometres per hour after 15 seconds. Find its acceleration.
Solution
Data:
From the question above
Initial velocity, u = 0 (because the motorcycle starts from rest)
Final velocity, v = speed = 108 km/h = (108 x 1000m) / (60 x 60s) = 108,000/3,600 = 30m/s
Time taken, t = 15 seconds
Therefore, we can now apply the formula that says
Acceleration = change in velocity/ time = (v-u)/t = (30-0)/15 =30/15 = 2ms-2
Therefore, the acceleration of the motorcycle is 2ms-2
Problem 4
A bus covers 50 kilometres in 1 hour. What is it is the average speed?
Solution
Data
Total distance covered = 50 km = 50 x 1000m = 50,000m
Time taken = 1 hour = 1 x 60 x 60s =3,600s
Average speed = Total distance covered by the object / Total time taken
Therefore, we can now calculate the average speed of the bus by substituting our data into the above formula
Hence,
Average speed = 50,000/3,600 = 13.9 m/s
Therefore, the average speed of the bus is 13.9m/s or approximately 14 meters per second.
Problem 5
A car travels 80 km in 1 hour and then another 20 km in the next hour. Find the average velocity of the car.
Solution
Data:
Initial displacement of the car = 80km
Final displacement of the car = 20km
The total displacement of the car = initial displacement of the car + final displacement of the car
Therefore,
The total displacement of the car = 80km + 20km = 100km
Also,
The time for 80km is 1hr
And the time for 20km is 1hr
Total time taken = The time for 80km (1hr) + The time for 20km (1hr)
Hence
Total time taken = 1hr + 1hr = 2hrs
Now, to calculate the average velocity of the car, we apply the formula that says
Average velocity = total displacement/total time taken = 100km/2hrs = 50km/h
We can further convert the above answer into meters per second
Average velocity = 50km/h = (50 x 1000m)/(1 x 60 x 60s) = 50,000/3,600 = 13.9m/s or 14ms-1
Therefore, the average velocity of the car is 14ms-1
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Problem 6
A body falls from the top of a tower 100 meters high and hits the ground in 5 seconds. Find its acceleration.
Solution
Data
Displacement = 100m
Time = 5 seconds
and we can apply the formula for acceleration that says
acceleration, a = Displacement/time2 = 100/52 = 100/25 = 4ms-2
Therefore the acceleration due to the gravity of the body is 4ms-2
Note: The acceleration due to the gravity of a body on the surface of the earth is constant (9.8ms-2), even though there may be a slight difference due to the mass and altitude of the body.
Problem 7
An object is thrown vertically upward at an initial velocity of 10ms-1 and reaches a maximum height of 50 meters. Find its initial upward acceleration.
Solution
Data
Initial velocity, u = 10ms-1
Final velocity, v = 0
maximum height = displacement = 50m
Initial upward acceleration, a =?
When we apply the formula that says a = (v2 – u2)/2d we will have
a = (0 – 102)/(2 x 50) = -100/100 = -1 ms-2
Hence, since our acceleration is negative, we can now say that we are dealing with retardation or deceleration.
Therefore, the retardation is -1ms-2
Note: Retardation is the negative of acceleration, thus it is written in negative form.
Problem 8
A car is traveling at a velocity of 8ms-1 and experiences an acceleration of 5ms-2. How far does it travel in 4 seconds?
Solution
Data:
Initial velocity, u = 8ms-1
acceleration, a = 5ms-2
Distance, s =?
time, t = 4s
We can apply the formula that says
s = ut + (1/2)at2
Thus
s = 8 x 4 + (1/2) x 5 x 42
s = 32 + (1/2) x 80 = 32 + 40 = 72m
Therefore, the distance covered by the car in 4s is 72 meters.
Problem 9
A body is traveling at a velocity of 10m/s and experiences a deceleration of 5ms-2. How long does it take the body to come to a complete stop?
Solution
Data
Initial velocity, u = 10m/s
Final velocity, v = 0
acceleration , a = retardation = -5ms-2
time, t = ?
We already know that acceleration, a = change in velocity/time
This implies that
Time, t = change in velocity/acceleration
Thus
t = (v – u)/t = (0-10)/-5 = -10/-5 = 2s
Therefore, the time it takes the car to stop is 2 seconds.
Problem 10
A body is traveling at a velocity of 20 m/s and has a mass of 10 kg. How much force is required to change its velocity by 10 m/s in 5 seconds?
Solution
Data
Change in velocity, v =10 m/s
mass of the object, m = 10kg
time, t = 5s
We can apply newton’s second law of motion which says f = ma
and since a = change in velocity/time
we will have an acceleration equal to
a = 10/5 = 2ms-2
Therefore, to find the force, we can now say
f = ma = 10 x 2 = 20N
Therefore, the force that can help us to change the velocity by 10 m/s in 5 seconds is 20-Newton.
Drop a comment if there is anything you don’t understand about speed, velocity, and acceleration Problems.
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