Differentiate position to get velocity, and velocity to get acceleration.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Speed is not velocity
Velocity carries a sign for direction. Speed is its magnitude, and is never negative.
When is it slowing down
When velocity and acceleration have opposite signs. Not when acceleration is simply negative.
Related rates
Differentiate a relationship with respect to time, and the two rates appear connected.
Position, velocity, acceleration
Velocity is the derivative of position and acceleration the derivative of velocity. Speed is the absolute value of velocity, and the distinction between speed and velocity is examined explicitly.
When is the particle speeding up?
Speed increases when velocity and acceleration share a sign, and decreases when they have opposite signs. This is a standard exam question and it is answered by comparing signs, not by differentiating speed.
Related rates: differentiate before substituting
Write the relationship, differentiate with respect to time, then substitute the known values. Substituting first freezes the variable you needed to vary, and it is the classic error in these problems.
Signs and units carry meaning
A shrinking quantity has a negative rate. The answer needs units — cubic centimetres per second, not just a number — and a sign consistent with the situation described.
Step 2: Try It Yourself
Tap and try it out.
- Point(1, -3)
- Slope of the tangent-1
Step 3: Watch an Example
One step at a time.
Watch Nadia Decide About Slowing Down
At t = 3, velocity is −4 and acceleration is −2.
- Step 1
The velocity is negative, so the object is moving backwards.
Step 4: Your Turn
Practice makes it stick.
Speed
Problem 1 of 2
Velocity is −7. What is the speed?
At Rest
Problem 2 of 2
An object is momentarily at rest. What is its velocity?
Reading Motion
1 of 8
Velocity 5, acceleration 2. Speeding up? 1 for yes, 0 for no.
2 of 8
Velocity 5, acceleration −2. Speeding up? 1 for yes, 0 for no.
3 of 8
Velocity −6, acceleration −3. Speeding up? 1 for yes, 0 for no.
4 of 8
Order the steps of a related-rates problem.
- 1Differentiate both sides with respect to time
- 2Substitute the known values and rates
- 3Solve for the unknown rate
- 4Write a relationship between the quantities
5 of 8
Position s(t) = t². What is the velocity at t = 3?
6 of 8
Same position function. What is the acceleration?
7 of 8
Velocity −3. Is the object moving forwards? 1 for yes, 0 for no.
8 of 8
Speed is 9 and the object moves backwards. What is the velocity?
Step 5: Quick Check
Show what you know.
Question 1 of 1
Velocity −4, acceleration 6. Speeding up? 1 for yes, 0 for no.
What You Learned
- Position differentiates to velocity, and velocity to acceleration.
- Speed is the magnitude of velocity and is never negative.
- An object speeds up when velocity and acceleration share a sign.