Velocity and Acceleration Calculator

This velocity and acceleration calculator takes a position function ( s(t) ) and a moment in time, and returns the velocity ( v(t) = s'(t) ), the acceleration ( a(t) = s”(t) ), their values at that moment, and whether the object is speeding up or slowing down.

It’s aimed at AP Calculus and physics students working through straight-line motion (“particle moving along a line”) problems. Also, it’s free, runs in your browser and needs no sign-up.

How to use the velocity and acceleration calculator

1. Type the position function into s(t) using the variable t. Use ^ for powers (t^3), and sin(), cos(), sqrt(), ln() and e^(t) as needed. Also, implicit multiplication works: 9t^2 and t sin(t) are fine.
2. Enter the time you care about in At time t. Values like pi/2 are accepted.
3. Press Analyze motion. You’ll see the formulas for ( v(t) ) and ( a(t) ), then ( s ), ( v ) and ( a ) at your time, plus a one-line verdict: speeding up, slowing down, momentarily at rest, or constant velocity at that instant.

How it works

Velocity is the rate of change of position and acceleration is the rate of change of velocity:

$$v(t) = s'(t), qquad a(t) = v'(t) = s''(t)$$

The calculator differentiates ( s(t) ) symbolically, twice, and evaluates all three functions at your ( t ). For polynomials, that’s just the power rule applied term by term.

The speeding-up test compares signs. Speed is ( |v| ), and it grows when velocity and acceleration point the same way:

• ( v ) and ( a ) have the same sign: speeding up.
• When ( v ) and ( a ) have opposite signs: slowing down.
• If ( v = 0 ): the object is momentarily at rest (a possible turnaround).

Worked example

A particle moves along a line. Its position is ( s(t) = t^3 – 9t^2 + 24t ) meters. Time ( t ) is in seconds. Describe its motion at ( t = 5 ) and at ( t = 1 ). You can use a velocity and acceleration calculator. For example, enter the same function in the calculator above.

Derivatives:

$$v(t) = 3t^2 - 18t + 24, qquad a(t) = 6t - 18$$

At ( t = 5 ): ( s(5) = 125 – 225 + 120 = 20 ) m, ( v(5) = 75 – 90 + 24 = 9 ) m/s, ( a(5) = 12 ) m/s². Both are positive, so the particle is moving right and speeding up.

At ( t = 1 ): ( s(1) = 16 ) m, ( v(1) = 9 ) m/s, ( a(1) = -12 ) m/s². Velocity is positive but acceleration is negative, so the particle is moving right while slowing down.

When is it at rest? Factor the velocity: ( v(t) = 3(t – 2)(t – 4) ). So ( v = 0 ) at ( t = 2 ) and ( t = 4 ). These are ( s )’s critical points. The particle can change direction here. It moves right until ( t = 2 ), reaching ( s = 20 ). Then it moves left until ( t = 4 ), returning to ( s = 16 ). After that, it moves right again. This is what a velocity and acceleration calculator shows.

Displacement vs total distance

Exams love this follow-up, and the calculator won’t do it for you, so here is how. Over ( 0 le t le 5 ):

• Displacement is ( s(5) – s(0) = 20 – 0 = 20 ) m. Equivalently, ( int_0^5 v(t),dt = 20 ) by the Fundamental Theorem of Calculus.
• Total distance adds up every leg without cancellation: ( 20 + 4 + 4 = 28 ) m, which equals ( int_0^5 |v(t)|,dt ). The definite integral calculator can evaluate that integral if you type the velocity with x in place of t inside abs().

First, use the turnaround times from ( v(t) = 0 ) to split the interval. Next, use the calculator at each endpoint and turnaround time to read off the positions.

Common mistakes

• Reading “negative acceleration” as “slowing down.” At ( t = 3.5 ), ( a = 3 > 0 ) but ( v = -2.25 < 0 ), so the particle slows down even though ( a ) is positive.
• Forgetting the units. If ( s ) is in meters and ( t ) in seconds, ( v ) is in m/s and ( a ) in m/s².
• Using ( x ) instead of ( t ). The input expects the variable t.

Speeding up or slowing down?

An object speeds up when velocity and acceleration have the same sign and slows down when they have opposite signs. For \( s(t) = t^3 - 6t^2 + 9t \):

• \( v(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3) \), zero at \( t = 1 \) and \( t = 3 \).
• \( a(t) = 6t - 12 \), zero at \( t = 2 \).

Interval	\( v \)	\( a \)	Motion
\( 0 < t < 1 \)	+	−	moving forward, slowing down
\( 1 < t < 2 \)	−	−	moving backward, speeding up
\( 2 < t < 3 \)	−	+	moving backward, slowing down
\( t > 3 \)	+	+	moving forward, speeding up

Going the other way: from acceleration to position

Integrate to work backwards. Use starting values for constants. If \( a(t) = -9.81 \), \( v(0) = 20 \) and \( s(0) = 0 \), then \( v(t) = 20 - 9.81t \) and \( s(t) = 20t - 4.905t^2 \). This is how the kinematic equations are derived for a velocity and acceleration calculator.

Practice problems

1. \( s(t) = 4t - t^2 \): find the velocity and acceleration at \( t = 1 \).
2. When is that object at rest?

Answers: 1. \( v(1) = 4 - 2 = 2 \), \( a = -2 \). 2. \( v(t) = 4 - 2t = 0 \) at \( t = 2 \).

Interpreting the Calculator's Output

After pressing 'Analyze motion', the velocity and acceleration calculator displays several key pieces of information. First, it shows the derived formulas for velocity v(t) and acceleration a(t). Always check these against your own calculations to ensure the input function was correctly interpreted. For example, if you entered t^2, expect v(t) = 2t.

Next, find numerical values for position s, velocity v, and acceleration a. These show at your specified time t. The sign of velocity tells the motion direction. Positive means moving in the positive direction. Negative means moving in the negative direction. Zero velocity means the object is momentarily at rest. This data is from the velocity and acceleration calculator.

Finally, the calculator provides a verdict on whether the object is speeding up or slowing down. This is determined by comparing the signs of velocity and acceleration. If both have the same sign (both positive or both negative), the object is speeding up. If they have opposite signs, it is slowing down. This instant check is very helpful.

Practical Tips for Using the Calculator

Always double-check your input function for typos. For example, a missing parenthesis or an incorrect operator can lead to unexpected results. Use ^ for exponents, like t^2 for t squared.

Remember that the calculator expects 't' as the variable. Using 'x' or any other letter will cause an error or misinterpretation. Also, be mindful of units; ensure consistency in your problem before and after using the calculator.

For trigonometric functions, ensure your calculator (or your understanding) is set to radians, as is standard in calculus. Pi is often used in time inputs for these functions. Finally, if you encounter 'momentarily at rest,' it's crucial to check the acceleration at that point to see if the object will start moving again or remain at rest.

Related guides

• Power Rule for Derivatives
• How to Find Critical Points
• Fundamental Theorem of Calculus
• Related Rates

Further reading

• Tangent Lines and Rates of Change (Paul’s Online Math Notes) — velocity as the instantaneous rate of change of position.
• Higher Order Derivatives (Paul’s Online Math Notes) — second derivatives, including acceleration as ( s”(t) ).

For definitions and formulas of velocity, acceleration and displacement, with motion graphs, read velocity and acceleration. For constant acceleration, the kinematic equations calculator solves SUVAT problems directly. Also, the projectile motion calculator handles launches at an angle.

FAQ

How do you find velocity from a position function?

Differentiate it: ( v(t) = s'(t) ). First, enter ( s(t) ) in the calculator and it shows ( v(t) ) as a formula and its value at your chosen time.

How do you find acceleration from position?

First, differentiate twice: ( a(t) = s”(t) ). The calculator displays ( a(t) ) and evaluates it at your time.

How can I tell if an object is speeding up?

Check the signs of ( v ) and ( a ) at that instant. Same sign means speeding up; opposite signs mean slowing down. The calculator prints this verdict for you.

Can it go from acceleration back to position?

No. However, it only works forward, from ( s(t) ) to ( v(t) ) and ( a(t) ). Going backward requires integration plus initial conditions; the integral calculator handles the antiderivative step.