Worksheet on calculating acceleration and time in physics problems.

Problem Analysis:

The task involves solving problems related to acceleration and time. Acceleration is defined as the rate of change of velocity with respect to time, and it can be calculated using the formula:

\[
a = \frac{\Delta v}{\Delta t} = \frac{v_f - v_i}{t}
\]

Where:
- \( a \) is acceleration,
- \( v_f \) is the final velocity,
- \( v_i \) is the initial velocity,
- \( t \) is the time interval.

We will solve each problem step by step.

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1. Write down the acceleration equation.

The acceleration equation is:

\[
a = \frac{v_f - v_i}{t}
\]

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2. A man running at 5 m/s increases his velocity to 10 m/s in 2 seconds. What is his acceleration?

#### Given:
- Initial velocity, \( v_i = 5 \, \text{m/s} \)
- Final velocity, \( v_f = 10 \, \text{m/s} \)
- Time, \( t = 2 \, \text{s} \)

#### Solution:
Using the acceleration formula:

\[
a = \frac{v_f - v_i}{t} = \frac{10 - 5}{2} = \frac{5}{2} = 2.5 \, \text{m/s}^2
\]

#### Answer:
\[
\boxed{2.5 \, \text{m/s}^2}
\]

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3. A car accelerates from rest to a speed of 50 m/s in 4 seconds. What was its acceleration?

#### Given:
- Initial velocity, \( v_i = 0 \, \text{m/s} \) (since it starts from rest)
- Final velocity, \( v_f = 50 \, \text{m/s} \)
- Time, \( t = 4 \, \text{s} \)

#### Solution:
Using the acceleration formula:

\[
a = \frac{v_f - v_i}{t} = \frac{50 - 0}{4} = \frac{50}{4} = 12.5 \, \text{m/s}^2
\]

#### Answer:
\[
\boxed{12.5 \, \text{m/s}^2}
\]

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4. A train travelling at 100 m/s slows down and stops at a platform in 25 seconds. What was its acceleration?

#### Given:
- Initial velocity, \( v_i = 100 \, \text{m/s} \)
- Final velocity, \( v_f = 0 \, \text{m/s} \) (since it stops)
- Time, \( t = 25 \, \text{s} \)

#### Solution:
Using the acceleration formula:

\[
a = \frac{v_f - v_i}{t} = \frac{0 - 100}{25} = \frac{-100}{25} = -4 \, \text{m/s}^2
\]

The negative sign indicates deceleration (slowing down).

#### Answer:
\[
\boxed{-4 \, \text{m/s}^2}
\]

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5. Extension: A car travels from 0 to 20 km/h in 0.5 seconds. What is its acceleration in m/s²?

#### Given:
- Initial velocity, \( v_i = 0 \, \text{km/h} \)
- Final velocity, \( v_f = 20 \, \text{km/h} \)
- Time, \( t = 0.5 \, \text{s} \)

#### Step 1: Convert velocities from km/h to m/s.
\[
1 \, \text{km/h} = \frac{1000}{3600} \, \text{m/s} = \frac{5}{18} \, \text{m/s}
\]
\[
v_f = 20 \, \text{km/h} = 20 \times \frac{5}{18} = \frac{100}{18} = \frac{50}{9} \, \text{m/s}
\]

#### Step 2: Use the acceleration formula.
\[
a = \frac{v_f - v_i}{t} = \frac{\frac{50}{9} - 0}{0.5} = \frac{\frac{50}{9}}{0.5} = \frac{50}{9} \times 2 = \frac{100}{9} \, \text{m/s}^2
\]

#### Answer:
\[
\boxed{\frac{100}{9} \, \text{m/s}^2 \, \text{or approximately } 11.11 \, \text{m/s}^2}
\]

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6. Write down the acceleration equation, rearranged for time.

The acceleration equation is:

\[
a = \frac{v_f - v_i}{t}
\]

Rearranging for time \( t \):

\[
t = \frac{v_f - v_i}{a}
\]

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7. A rocket is flying at 400 m/s and accelerates at 20 m/s² until it reaches its top speed of 1000 m/s. How much time does this take?

#### Given:
- Initial velocity, \( v_i = 400 \, \text{m/s} \)
- Final velocity, \( v_f = 1000 \, \text{m/s} \)
- Acceleration, \( a = 20 \, \text{m/s}^2 \)

#### Solution:
Using the rearranged time formula:

\[
t = \frac{v_f - v_i}{a} = \frac{1000 - 400}{20} = \frac{600}{20} = 30 \, \text{s}
\]

#### Answer:
\[
\boxed{30 \, \text{s}}
\]

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Final Answers:

1. \( a = \frac{v_f - v_i}{t} \)
2. \( \boxed{2.5 \, \text{m/s}^2} \)
3. \( \boxed{12.5 \, \text{m/s}^2} \)
4. \( \boxed{-4 \, \text{m/s}^2} \)
5. \( \boxed{\frac{100}{9} \, \text{m/s}^2 \, \text{or approximately } 11.11 \, \text{m/s}^2} \)
6. \( t = \frac{v_f - v_i}{a} \)
7. \( \boxed{30 \, \text{s}} \)