πAcceleration
What is Acceleration?
Acceleration is the rate of change of velocity. Just as velocity tells us how fast position is changing, acceleration tells us how fast velocity is changing.
a_avg = Ξv / Ξt
Acceleration is a vector β it has direction. If you're moving right and speeding up, acceleration points right. If you're moving right but slowing down, acceleration points left (opposing motion).
Key insight: acceleration doesn't have to be in the same direction as velocity! A ball thrown upward has downward acceleration (due to gravity) even while it's still moving upward.
Think About It
A car is traveling at 20 m/s north. It then turns and travels at 20 m/s east. Has the car accelerated? (Speed didn't change!)
Interpreting Velocity-Time Graphs
On a velocity-time (v-t) graph: - Slope = acceleration - Area under curve = displacement
Positive slope β positive acceleration (speeding up in + direction) Negative slope β negative acceleration (slowing down or speeding up in β direction) Zero slope (horizontal line) β constant velocity, zero acceleration
If the v-t graph is a straight line, acceleration is constant β this is the most common case on the AP exam.
Velocity-Time Graph
Adjust the acceleration slider and watch how the velocity-time graph changes. Notice the area under the graph gives displacement.
βοΈ Worked Example
Problem: A car starts from rest and reaches 30 m/s in 6 seconds. What is its average acceleration?
β οΈ Common Mistakes
Misconception: Negative acceleration always means the object is slowing down.
β Correct thinking: Negative acceleration means acceleration points in the negative direction. If the object is also moving in the negative direction, it is actually speeding up.
Why: Deceleration depends on the relationship between the signs of velocity and acceleration, not on the sign of acceleration alone.
Misconception: An object with zero velocity must have zero acceleration.
β Correct thinking: An object can have zero instantaneous velocity and non-zero acceleration simultaneously. A ball at the peak of its throw has v = 0 but a = β9.8 m/sΒ² (gravity still acts).
Why: Velocity and acceleration are independent quantities. Acceleration is the rate of change of velocity, not the velocity itself.
Misconception: The area under a position-time graph gives displacement.
β Correct thinking: The slope of a position-time graph gives velocity. The area under a velocity-time graph gives displacement.
Why: Mixing up which graph operation corresponds to which kinematic quantity is one of the most common AP exam errors.
π Practice Problems
Try these problems. Check your answer when ready.
A car accelerates from 0 to 24 m/s in 8 seconds. What is its average acceleration?
a = (Ξ v)/(Ξ t)A skateboard rolls at 6 m/s and comes to rest in 3 seconds due to friction. What is the acceleration?
A velocity-time graph shows a straight line from (0 s, 10 m/s) to (5 s, β10 m/s). What is the acceleration? What is the displacement over those 5 seconds?
A ball is thrown straight up with initial velocity 20 m/s. Using a = β10 m/sΒ², find (a) the time to reach the peak, and (b) the velocity at t = 3 s.
v = v_0 + atSketch a velocity-time graph for the following motion: starts at rest, accelerates uniformly for 4 s to 12 m/s, maintains constant speed for 2 s, then decelerates uniformly back to rest in 3 s. What is the total displacement?
Finished reading through this lesson?