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Speed-time graph, also known as the velocity-time graph. In this graph time is marked on the vertical x-axis and speed is marked on the horizontal y-axis. When the graph drawn is a straight line on the y-axis of the graph it denotes that speed is constant. Speed is the distance that an object travels in relation to the amount of time it takes to travel. A time standard is a specification for measuring the time that establishes a chronology, quantifies the length of a time interval, and assigns a number or calendar date to an instant (point in time) (ordering of events).
Read More: Motion in a Straight Line
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Key Terms: X-axis, Y-axis, Speed, Time, Distance-Time Graph, Acceleration, Rate of Change of Speed, Velocity, Horizontal line
Speed Time Graph
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In Speed-time graph, speed is always indicated on the Y-axis or vertical axis, and time is always shown on the horizontal, or X-axis.
where,
u = the initial speed
v = stands for the end speed.
t = stands for time.
At rest, the object’s initial speed is zero.
Read More: Horizontal and Vertical Lines
Example: Consider a car that accelerates from a standstill to 15 mph in ten seconds. It then travels for another 10 seconds at the same consistent speed. It accelerates to 25 m/s for additional 10 seconds after the 20th second. When the car notices a rabbit darting across the road, it applies the brakes and comes to a complete stop in 20 seconds. As a result, its final speed is 0.
Known: Car is going at a consistent pace between the 10th and 20th seconds, resulting in a horizontal line as speed is constant
The vehicle accelerates from 15 m/s to 25 m/s in ten seconds.
Compute the gradients at various locations in this graph.
The acceleration, or rate of change of speed, is obtained by calculating the gradients in this graph.
There is no acceleration when a body travels at a constant speed. The car is driving at a constant speed between the 10th and 20th seconds, hence there is no acceleration or the acceleration is 0m/s². When calculating acceleration, the formula is always final speed – beginning speed divided by time.
As a result, the vehicle accelerates from 0 to 15 m/s in the first ten seconds. The end speed is 15 mph, and the initial speed is 0 mph, all in 10 seconds.
To find the acceleration for this segment,
Multiply 15 – 0 = 15/10, which is 1.5m/s2.
Take the last segment, where the vehicle had to apply brakes due to the rabbit getting in the way, and it came to a complete stop.
The ultimate velocity is 0 and the initial velocity is 25m/s in this case.
Difference in speed is 0-25 = -25m/s, and it takes 20 seconds.
As a result, the acceleration is -25/20, or -1.25 m/s2.
- The "Negative" sign denotes the reversal of acceleration, i.e., a reduction in speed, also known as deceleration.
- The distance travelled can be calculated by calculating the area under the graph.
- Make squares, rectangles, or triangles out of the area beneath the graph.
- Calculate the area of each one and combine them to get the total distance.
- The units for distance and time might be km and hours, respectively, and the speed will be expressed in km/hr.
- The magnitude of the velocity at a given time is the speed at that time.
- As a result, equations of motion can be used to create speed-time graphs.
Motion in a Straight Line Video Explanation
Read More: Line Graph
Speed Time Graphs with Constant Speed
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The speed-time graph, with the Y-axis signifying speed and the X-axis denoting time, will look like this
when the speed is constant:
The speed is constant (c) throughout the time span, as seen in the graph.
The speed will always be c, regardless of how much time passes.
Example: If a particle's acceleration is zero (0) and its speed is constant, say 5 m/s at t = 0, the particle's speed will remain constant throughout time.
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Speed Time Graphs with Constant Acceleration
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When the particle's initial speed is zero and the acceleration is constant, the particle's speed will grow linearly, as predicted by the equation:
| v = u + at |
Since u = 0
| v = at |
The particle's speed will rise linearly with respect to time, as indicated in the diagram. The magnitude of acceleration will be determined by the graph's slope.
When a particle's acceleration is constant (k) and its initial speed is zero, the particle's speed increases or decreases linearly.
The magnitude of acceleration will be determined by the slope of the speed-time graph (k).
Read More: Air resistance formula
Speed Time Graph with Increasing Acceleration
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The speed-time graph will be a curve when the acceleration increases with time, as predicted by the equation:
| v = u + at |
Since u = 0
| v= at |
The speed-time graph will be a curve because acceleration is a function of time.
Note that because the acceleration is growing over time, the amplitude of the slope will likewise be increasing over time.
Speed Time Graph with Increasing Acceleration
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The speed-time graph will be a curve if the acceleration of a particle is a function of time and the initial speed is zero. The magnitude of acceleration at any instant (at a given time) is determined by the slope of the speed-time graph.
When the acceleration decreases with time, the graph will be different.
Read More: Difference Between Acceleration and Velocity
Acceleration and Its Units
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Acceleration is the pace at which a person's speed varies. Acceleration in physics refers to a speed increase. When a vehicle advertisement reads "0 to 100 in 5 seconds," it's usually referring to acceleration.
- There is also another aspect of acceleration generated by a direction change.
- Even though there is no change in speed, there is acceleration if the direction changes. Since velocity is a vector quantity.
- Any vector quantity has a magnitude and a direction; hence, acceleration exists even if the value does not change but the direction does.
For Example: In a circular motion, there is a force termed centripetal acceleration. Acceleration has immediate and average acceleration, just like speed. Instantaneous acceleration is acceleration measured over a very brief time interval.
Change in speed (m/s) divided by time (s) equals acceleration, which is expressed in m/s2.
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Things to Remember
- Speed-time graph is also called the velocity-time graph.
- Acceleration is the pace at which the speed varies.
- Speed describes how quickly or slowly an object moves.
- Motion refers to a change in the location of an object over a period of time.
- Speed refers to the amount of distance travelled in a given amount of time.
- The initial speed of the object is zero at rest.
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Sample Questions
Ques. The speed-time graph shows a 50-second car journey, find which section of the graph has the greatest acceleration. (5 Marks)
Ans. We know,
The gradient of the line = Acceleration
We must find the gradient of each section.
Section A: Acceleration between 0s and 10s = {15-0}/ {10-0}=1.5 m/s2
Section B: This section is flat, meaning the acceleration will be 0
Section C: Acceleration between 20s and 30s = {25-15}/ {30-20}=1 m/s2
Section D: Acceleration between 30s and 50s = {0-25}/ {50-30}=−1.25 m/s2
Section A has the largest acceleration, so the maximum acceleration is 1.5 m/s2
Note: units of acceleration are expressed in distance/time2, which in this case is m/s2
Ques. The speed-time graph shows a 50-second car journey, Calculate the total distance travelled over the 50 seconds. (5 Marks)
Ans. We know,
Area under the graph = Distance travelled
To work out the area under this graph, we will break it into 4 shapes: A, B, C, and D.
This gives two triangles, a rectangle, and a trapezium, which are all shapes that we can work out the area of.
A= ½ ×10×15=75 m
B=10×15=150 m
C=½ ×(15+25)×10=200 m
D=½ ×20×25=250 m
Total distance travelled:
75+150+200+250=675 m
Ques. The below Figure represents the speed time graph for a particle. Find the distance covered by the particle (5 Marks)
(a) between t= 30min. and t= 40min.
(b) between t=10 min and t=40 min
(c) what is the speed of the particle?
Ans. a. Time taken =10 min = 10/60 hr
Speed of the graph =15 km/hr
Distance =Speed X time taken =15 x 10/60= 2.5 Km
- Time taken =30 min = 30/60 hr
Speed of the graph =15 km/hr
Distance =Speed X time taken =15 x 30/60= 7.5 Km
- 15 km/hr
Ques. A particle starts from rest and moves with a uniform acceleration of 5m/s2 for 10s and then it moves with a constant velocity for 4s. Later it slows down and comes to rest in 5s. Draw the velocity graph for the motion of the body and answer the following questions:
(a) What is the maximum velocity attended by the body?
(b) What is the distance travelled during this period of acceleration?
(c) What is distance travelled when the body was moving with constant velocity?
(d) What is the retardation of the body while slowing down?
(e) What is the distance travelled by retarding?
(f) What is the total distance travelled? (5 Marks)
Ans. (a) Maximum velocity will be reached when acceleration is stopped at the end of first 10 sec and it is shown clearly in graph
v=50 m/s
(b) Distance is given by the area of graph enclosed till 10 sec
s= (1/2)X 50 X10 = 250 m
(c) distance travelled when the body was moving with constant velocity is given by the enclosed graph from A to B
= 4X50 =200m
(d)Retardation is given by the slope of the curve from B to C
a=(0-50)/5= -10m/s2
(e) Distance = (1/2) X 50 X 5 =125 m
(f) Total distance is given by the total area of the graph = 250+200+125=575m
Ques. What does a horizontal line on a speed-time graph indicate? (1 Mark)
(a) Constant distance
(b) Constant speed
(c) Constant acceleration
Ans. a. constant distance.
Ques. What is the area under a speed-time graph equal to? (1 Mark)
(a) Acceleration
(b) Instantaneous speed
(c) Distance travelled
Ans. c. Distance travelled
Ques. What is the slope of a line on a velocity time graph an indication of? (1 Mark)
(a) The distance travelled by an object
(b) The instantaneous speed of an object
(c) The acceleration of an object
Ans. c. Acceleration of an object
Ques. What are the different types of speed-time graphs? Draw the graph. (3 Marks)
Ans. Three types of speed time graphs are:
- Speed time graph with constant speed
- Speed time graph with constant acceleration
- Speed time graph with increasing acceleration
Ques. 
Which of the following scenarios has caused the motion described by the graph?
(a) An object rolling down a slope then rolling up a slope
(b) An object rolling down a steep slope
(c) An object rolling up a slope then returning to its original position (1 Mark)
Ans. b, an object rolling down a steep slope.
Ques. Two objects are thrown vertically upwards simultaneously with their velocities x and y respectively. Prove that
(a) The heights reached by them would be in the ratio of x2: y2
(b) The time taken to reach the maximum height would be in the ratio of x:y
(Assume upward acceleration is –g and downward acceleration to be +g). (4 Marks)
Ans. Third equation of motion: v² = u²+2as
Here u = x, a = -g and v = 0
0 = x2-2gs
s1 = x22g
In the same manner
u = y, a = -g and v = 0
0 = y2-2gs
s2 = y2/2g
Hence ratio of height: s1/s2 = x2:y2
Deriving from the 1st equation of motion
v = u + at
0 = x - gt1
Or t1 = x/g
In the same manner
t2 = y/g
t1/t2 = x:y
Ques. A car moves at a speed of 40km/h. It is stopped by applying brakes which produces a uniform acceleration of -0.6m/s2. How much distance will the vehicle move before coming to stop? (2 Mark)
Ans. u = 40 km/h = 11.11m/s
A = -0.6m/s2
v = 0
v2 = u2 + 2as
s = 102.87 m
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