Physics Notes - Chapter 2.1: Acceleration and Average Speed
Acceleration
Acceleration is defined as the rate of change of speed over time.
It quantifies how much the speed changes within a specific time interval.
Formula: a = \frac{\Delta V}{\Delta T}, where:
- a represents acceleration (measured in meters per second squared, m/s²).
- \Delta V denotes the change in velocity (measured in meters per second, m/s).
- \Delta T signifies the change in time (measured in seconds, s).
\Delta V is calculated as the difference between the final velocity (Vf) and the initial velocity (Vi): \Delta V = Vf - Vi
\Delta T is calculated as the difference between the final time (T2) and the initial time (T1): \Delta T = T2 - T1
Important Note: The formula for acceleration is a = \frac{\Delta S}{\Delta T}, not a = \frac{S}{T}. The reason is that acceleration is based on the change in speed ([\Delta S]).
- Example: If a car maintains a constant speed of 60 m/s for 20 seconds, the speed (S) is 60 m/s, but the change in speed ([\Delta S]) is 0 m/s, resulting in no acceleration.
Example: Calculating Acceleration
Problem: A plane starts from rest and reaches a speed of 200 m/s in 10 seconds. Calculate the acceleration.
Step 1: Identify the variables.
- Initial speed (S_1) = 0 m/s (starts at rest)
- Final speed (S_2) = 200 m/s
- Change in time (\Delta T) = 10 seconds
- Acceleration (a) = ?
Step 2: Calculate the change in speed (\Delta S).
- \Delta S = S2 - S1 = 200 \text{ m/s} - 0 \text{ m/s} = 200 \text{ m/s}
Step 3: Apply the formula for acceleration.
- a = \frac{\Delta S}{\Delta T}
Step 4: Solve for acceleration.
- a = \frac{200 \text{ m/s}}{10 \text{ s}} = 20 \text{ m/s}^2
Deceleration (Negative Acceleration)
Deceleration occurs when an object slows down.
It is indicated by a negative sign in the acceleration value.
Example: Calculating Deceleration
Problem: A race car initially traveling at 400 m/s comes to a complete stop in 20 seconds. Calculate the car’s acceleration.
Step 1: Identify the variables.
- Initial speed (S_1) = 400 m/s
- Final speed (S_2) = 0 m/s (stops)
- Change in time (\Delta T) = 20 seconds
- Acceleration (a) = ?
Step 2: Calculate the change in speed (\Delta S).
\Delta S = S2 - S1 = 0 \text{ m/s} - 400 \text{ m/s} = -400 \text{ m/s}
Note the negative sign, indicating deceleration.
Step 3: Apply the formula for acceleration.
- a = \frac{\Delta S}{\Delta T}
Step 4: Solve for acceleration.
- a = \frac{-400 \text{ m/s}}{20 \text{ s}} = -20 \text{ m/s}^2
Average Speed
Average speed is calculated by dividing the total distance traveled by the total time taken.
Formula: S{ave} = \frac{D{total}}{T_{total}}, where:
- S_{ave} represents average speed.
- D_{total} denotes the total distance.
- T_{total} signifies the total time.
Example: Calculating Average Speed
Problem: A person walks 4 miles in 2 hours, rests for 1 hour, and then walks 8 miles in 3 hours. Calculate the person’s average speed.
Distance and Time:
- Segment 1: 4 miles in 2 hours
- Segment 2: 0 miles in 1 hour (rest)
- Segment 3: 8 miles in 3 hours
Totals:
- Total distance (D_{total}) = 4 miles + 0 miles + 8 miles = 12 miles
- Total time (T_{total}) = 2 hours + 1 hour + 3 hours = 6 hours
Calculation:
- S_{ave} = \frac{12 \text{ miles}}{6 \text{ hours}} = 2 \text{ mph}
Speed vs. Velocity
Speed is a scalar quantity, indicating how fast an object is moving without regard to direction.
Velocity is a vector quantity, specifying both the speed and direction of an object’s motion.
Examples:
- Speed: A person walks 4 m/s.
- Velocity: A person walks 2 m/s north.
- Velocity: A car drives 60 mph toward Dallas.
- Speed: A car drives 30 mph.
Scalars vs. Vectors
Scalar: A quantity that has magnitude only.
- Example: A boat is pulled by a 53 newton force (scalar).
Vector: A quantity that has both magnitude and direction.
- Example: A 14 newton force pull 30° left of north (vector).
Key Differences:
- Speed is a scalar (no direction).
- Velocity is a vector (has direction).
Vectors are defined by both magnitude and direction.
Practice Problems
Matching Variables with Quantities:
- a = 3 m/s²
- S or v = 23 meters/sec
- m = 23 kilograms
- D = 23 meters
- F = 23 newtons
- T = 23 sec
Acceleration Problems:
- Problem 1: A plane stops from 250 mph in 25 seconds. Calculate the plane’s acceleration.
- Problem 2: A person starts running from 2 m/s to 6 m/s in 2 seconds. Calculate the person’s acceleration.
- Problem 3: A dragster’s top acceleration is 60 m/s². If it accelerates for 3 seconds from the starting line, how fast will it be going?
Average Speed Problem:
- Problem: A guy bikes 15 miles in 1 hour, then rests for an hour. Then he bikes 25 miles in 2 hours. What was his average speed for the trip?
Identifying Scalars and Vectors
Speed (S) or Velocity (V):
- A bike goes 25 m/s toward main street. (V)
- A person walks 4 mph. (S)
- A plane flies 200 m/s. (S)
- A bird flies 100 mph due south. (V)
Scalar (S) or Vector (V):
- 40 mph toward Dallas. (V)
- A 25 N force pulls on a wagon. (V)
- 10 meters up the hill. (V)
- 12 meter per sec². (S)
Graph Analysis: Speed vs. Time
Graphing Speed vs. Time:
- The slope of a speed vs. time graph represents acceleration.
Interpreting Segments of a Speed vs. Time Graph:
- Constant speed: Horizontal line.
- Deceleration: Line with a negative slope.
- Accelerating: Line with a positive slope.
The slope of a position vs. time graph represents velocity, while the slope of a speed vs. time graph represents acceleration.