Speed Time Graphs Worksheet | Fun and Engaging PDF Worksheets - Free Printable

Let's solve each problem step by step using the provided speed-time graphs.

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Problem 1: Train Slowing Down

#### Graph Description:
The graph shows a straight line starting at 30 m/s and decreasing linearly to 0 m/s over 60 seconds.

#### Questions:
1. a) What is the deceleration of the train?
- Deceleration is the rate of change of speed with respect to time. It can be calculated as:
\[
\text{Deceleration} = \frac{\Delta \text{Speed}}{\Delta \text{Time}}
\]
From the graph:
- Initial speed (\(v_i\)) = 30 m/s
- Final speed (\(v_f\)) = 0 m/s
- Time interval (\(\Delta t\)) = 60 s
\[
\text{Deceleration} = \frac{0 - 30}{60} = -0.5 \, \text{m/s}^2
\]
The negative sign indicates deceleration.

Answer: \( \boxed{-0.5 \, \text{m/s}^2} \)

2. b) What is the distance travelled?
- The distance travelled is the area under the speed-time graph.
- The graph forms a right triangle with:
- Base (\(t\)) = 60 s
- Height (\(v\)) = 30 m/s
- Area of the triangle:
\[
\text{Distance} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 60 \times 30 = 900 \, \text{m}
\]

Answer: \( \boxed{900 \, \text{m}} \)

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Problem 2: Car's Journey

#### Graph Description:
The graph shows a car accelerating to 15 m/s, maintaining that speed for some time, and then decelerating back to 0 m/s over 60 seconds.

#### Questions:
1. a) What is the maximum speed of the car?
- The maximum speed is the highest point on the graph, which is 15 m/s.

Answer: \( \boxed{15 \, \text{m/s}} \)

2. b) What is the initial acceleration?
- Initial acceleration is the slope of the graph during the first segment (from 0 to 20 seconds).
- From the graph:
- Initial speed (\(v_i\)) = 0 m/s
- Final speed (\(v_f\)) = 15 m/s
- Time interval (\(\Delta t\)) = 20 s
\[
\text{Initial Acceleration} = \frac{15 - 0}{20} = 0.75 \, \text{m/s}^2
\]

Answer: \( \boxed{0.75 \, \text{m/s}^2} \)

3. c) What is the final deceleration?
- Final deceleration is the slope of the graph during the last segment (from 50 to 60 seconds).
- From the graph:
- Initial speed (\(v_i\)) = 15 m/s
- Final speed (\(v_f\)) = 0 m/s
- Time interval (\(\Delta t\)) = 10 s
\[
\text{Final Deceleration} = \frac{0 - 15}{10} = -1.5 \, \text{m/s}^2
\]

Answer: \( \boxed{-1.5 \, \text{m/s}^2} \)

4. d) What is the distance travelled?
- The distance is the area under the graph.
- The graph consists of three segments:
1. A triangle from 0 to 20 seconds:
\[
\text{Area}_1 = \frac{1}{2} \times 20 \times 15 = 150 \, \text{m}
\]
2. A rectangle from 20 to 50 seconds:
\[
\text{Area}_2 = 30 \times 15 = 450 \, \text{m}
\]
3. A triangle from 50 to 60 seconds:
\[
\text{Area}_3 = \frac{1}{2} \times 10 \times 15 = 75 \, \text{m}
\]
- Total distance:
\[
\text{Total Distance} = 150 + 450 + 75 = 675 \, \text{m}
\]

Answer: \( \boxed{675 \, \text{m}} \)

5. e) What is the average speed for the whole journey?
- Average speed is the total distance divided by the total time.
- Total distance = 675 m
- Total time = 60 s
\[
\text{Average Speed} = \frac{675}{60} = 11.25 \, \text{m/s}
\]

Answer: \( \boxed{11.25 \, \text{m/s}} \)

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Problem 3: Speedboat's Journey

#### Graph Description:
The graph shows the speedboat accelerating initially, maintaining a constant speed, and then decelerating.

#### Questions:
1. a) Find the acceleration between \( t = 0 \) and \( t = 2 \).
- Acceleration is the slope of the graph during this interval.
- From the graph:
- Initial speed (\(v_i\)) = 0 m/s
- Final speed (\(v_f\)) = 20 m/s
- Time interval (\(\Delta t\)) = 2 s
\[
\text{Acceleration} = \frac{20 - 0}{2} = 10 \, \text{m/s}^2
\]

Answer: \( \boxed{10 \, \text{m/s}^2} \)

2. b) Describe the speedboat’s motion between \( t = 2 \) and \( t = 6 \).
- Between \( t = 2 \) and \( t = 6 \):
- The speed remains constant at 20 m/s.
- This indicates uniform motion with zero acceleration.

Answer: The speedboat is moving at a constant speed of 20 m/s.

3. c) Find the distance travelled in the first 4 seconds.
- The distance is the area under the graph from \( t = 0 \) to \( t = 4 \).
- The graph consists of two segments:
1. A triangle from 0 to 2 seconds:
\[
\text{Area}_1 = \frac{1}{2} \times 2 \times 20 = 20 \, \text{m}
\]
2. A rectangle from 2 to 4 seconds:
\[
\text{Area}_2 = 2 \times 20 = 40 \, \text{m}
\]
- Total distance:
\[
\text{Total Distance} = 20 + 40 = 60 \, \text{m}
\]

Answer: \( \boxed{60 \, \text{m}} \)

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Problem 4: Car and Motorbike Journey

#### Graph Description:
The graph shows two lines representing the speeds of a car and a motorbike over time. Both are travelling in the same direction.

#### Questions:
1. a) When are they travelling at the same speed?
- They are travelling at the same speed when their speed values are equal, i.e., when the two lines intersect.
- From the graph, the intersection occurs at \( t = 20 \) seconds.

Answer: \( \boxed{20 \, \text{s}} \)

2. b) How far apart are they when they are travelling at the same speed?
- To find the distance between them at \( t = 20 \) seconds, we calculate the area under each graph from \( t = 0 \) to \( t = 20 \) and find the difference.
- Car:
- The graph is a triangle from 0 to 20 seconds:
\[
\text{Area}_{\text{Car}} = \frac{1}{2} \times 20 \times 30 = 300 \, \text{m}
\]
- Motorbike:
- The graph is a triangle from 0 to 10 seconds followed by a rectangle from 10 to 20 seconds:
\[
\text{Area}_{\text{Motorbike}} = \frac{1}{2} \times 10 \times 20 + 10 \times 20 = 100 + 200 = 300 \, \text{m}
\]
- Difference in distance:
\[
\text{Difference} = 300 - 300 = 0 \, \text{m}
\]

Answer: \( \boxed{0 \, \text{m}} \)

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Final Answers:

1. Train:
- a) \( \boxed{-0.5 \, \text{m/s}^2} \)
- b) \( \boxed{900 \, \text{m}} \)

2. Car:
- a) \( \boxed{15 \, \text{m/s}} \)
- b) \( \boxed{0.75 \, \text{m/s}^2} \)
- c) \( \boxed{-1.5 \, \text{m/s}^2} \)
- d) \( \boxed{675 \, \text{m}} \)
- e) \( \boxed{11.25 \, \text{m/s}} \)

3. Speedboat:
- a) \( \boxed{10 \, \text{m/s}^2} \)
- b) The speedboat is moving at a constant speed of 20 m/s.
- c) \( \boxed{60 \, \text{m}} \)

4. Car and Motorbike:
- a) \( \boxed{20 \, \text{s}} \)
- b) \( \boxed{0 \, \text{m}} \)