Instantaneous Speed Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Instantaneous Speed.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

Instantaneous speed is the speed of an object at a particular moment in time.

It is what a speedometer shows right now, not over the whole trip.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Instantaneous Speed starts by naming what changes, over what time interval, and whether direction matters.

Common stuck point: Students often know a formula related to instantaneous speed but skip the recognition step: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?

Worked Examples

Example 1

medium

Position is

x(t) = 3t^2

m. Find the instantaneous speed at

t = 4 \text{ s}

(slope

6t

Answer

24 m/s

First step

v(t) = dx/dt = 6t

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Example 2

medium

A ball falls from rest under gravity (

g=10 \text{ m/s}^2

). Find its instantaneous speed at

t = 2 \text{ s}

Example 3

medium

Position is

x(t) = 4t - t^2

m. Find the time when instantaneous speed is zero (slope

4 - 2t

Example 4

hard

A ball is thrown up at

25 \text{ m/s}

(

g=10 \text{ m/s}^2

). Find the instantaneous speed at

t = 1 \text{ s}

and at

t = 4 \text{ s}

Example 5

hard

A particle moves so that

x(t) = t^3 - 9t

m. Find the times when its instantaneous speed is zero (slope

3t^2 - 9

Example 6

hard

A particle's velocity-time graph is a triangle:

v=0

t=0

, peaks at

v=12 \text{ m/s}

t=3 \text{ s}

, drops linearly back to

v=0

t=6 \text{ s}

. Find its instantaneous speed at

t = 2 \text{ s}

Example 7

hard

x(t) = 2t + \sin(t)

m. Estimate the instantaneous speed at

t = 0

s using the slope

2 + \cos(t)

Example 8

challenge

A particle has

x(t) = t^2 - 4t + 3

m for

0 \le t \le 5

. Find both the time when instantaneous speed is zero and the maximum instantaneous speed on the interval.

Example 9

challenge

An object's position is given by

x(t) = 6t^2 - t^3

m for

t \ge 0

. Find the maximum instantaneous speed on

0 \le t \le 5 \text{ s}

(slope

12t - 3t^2

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A speedometer reads

25

m/s right now. What kind of speed is this?

Example 2

easy

Is instantaneous speed a scalar or a vector?

Example 3

easy

Position is

x(t)=4t

m. What is the instantaneous speed at

t=2

Example 4

easy

During uniform motion at

10

m/s, what is the instantaneous speed at any moment?

Example 5

easy

A car's speedometer shows

0

at a red light. What is its instantaneous speed?

Example 6

easy

Which describes a single moment: average speed or instantaneous speed?

Example 7

easy

A camera flash captures a ball at

12

m/s at one instant. Instantaneous or average speed?

Example 8

easy

Over a whole trip a car's average speed is

50

km/h. Must its instantaneous speed always be

50

km/h?

Example 9

medium

Position is

x(t)=t^2

m. Estimate the instantaneous speed at

t=3

s using the slope

2t

Example 10

medium

Between

t=2

s and

t=2.1

s, an object moves from

4.0

m to

4.5

m. Estimate the instantaneous speed near

t=2

Example 11

medium

A ball thrown up reaches its peak. What is its instantaneous speed there?

Example 12

medium

A car at constant acceleration goes from

0

20

m/s in

10

s. Its instantaneous speed at

t=5

Example 13

medium

A position-time graph is a straight line through the origin with slope

7

. What is the instantaneous speed?

Example 14

medium

Over

0

4

s a car's instantaneous speed rises steadily from

0

8

m/s. What is its instantaneous speed at

t=2

Example 15

challenge

Position is

x(t)=t^3

m. Find the instantaneous speed at

t=2

s (slope is

3t^2

Example 16

challenge

An object's average speed over

0

4

s is

6

m/s, yet its instantaneous speed at

t=4

s is

12

m/s. Is this consistent with starting from rest under constant acceleration?

Example 17

challenge

Position is

x(t)=5t-t^2

m. At what time is the instantaneous speed zero (slope

5-2t

Example 18

medium

Position is

x(t)=2t^2

m. Find the instantaneous speed at

t=3

s (slope

4t

Example 19

medium

Between

t=1

s and

t=1.01

s an object moves

0.0

0.0

... actually from

2.00

m to

2.06

m. Estimate the instantaneous speed near

t=1

Example 20

medium

A car from rest at constant

a=3

m/s

^2

. Find its instantaneous speed at

t=4

Example 21

easy

Position is

x(t)=8t

m. Find the instantaneous speed at

t=5

Example 22

easy

A speedometer reads

80 \text{ km/h}

right now. Is this instantaneous or average speed?

Example 23

easy

A ball thrown upward returns to the thrower with the same speed it had on the way up. Is its instantaneous speed at the highest point greater than, less than, or equal to the launch speed?

Example 24

easy

An object moves

20 \text{ m}

4 \text{ s}

at constant speed. Find its instantaneous speed at

t = 2 \text{ s}

Example 25

easy

A car at constant acceleration starts from rest and reaches

30 \text{ m/s}

6 \text{ s}

. Find its instantaneous speed at

t=3 \text{ s}

Example 26

medium

Between

t=5 \text{ s}

and

t=5.01 \text{ s}

a runner moves from

30.00 \text{ m}

30.07 \text{ m}

. Estimate the instantaneous speed near

t=5 \text{ s}

Example 27

medium

A car decelerates from

20 \text{ m/s}

at constant

a = -2 \text{ m/s}^2

. Find its instantaneous speed at

t = 6 \text{ s}

Example 28

medium

Position is

x(t) = 5t^2 - 2t

m. Find the instantaneous speed at

t = 1 \text{ s}

(slope

10t - 2

Example 29

medium

A position-time graph shows a curve whose slope at

t=4 \text{ s}

-3

(m per s). Find the instantaneous speed at

t=4 \text{ s}

Example 30

medium

Position is

x(t) = 100 - 5t^2

m. Find the instantaneous speed at

t=3 \text{ s}

(slope

-10t

Example 31

medium

Between

t = 2.00 \text{ s}

and

t = 2.05 \text{ s}

a glider moves from

10.00 \text{ m}

10.35 \text{ m}

. Estimate the instantaneous speed near

t = 2 \text{ s}

Example 32

medium

A bicycle accelerates from

2 \text{ m/s}

a = 1.5 \text{ m/s}^2

. Find its instantaneous speed at

t = 4 \text{ s}

Example 33

hard

Position is

x(t) = t^3 - 6t^2 + 9t

m. Find the instantaneous speed at

t = 4 \text{ s}

(slope

3t^2 - 12t + 9

Example 34

hard

A car decelerates uniformly from

30 \text{ m/s}

and stops in

6 \text{ s}

. Find its instantaneous speed at

t = 4 \text{ s}

Example 35

hard

A ball is dropped from

80 \text{ m}

(

g=10 \text{ m/s}^2

). Find its instantaneous speed just before it hits the ground.

Example 36

hard

A train moving at

30 \text{ m/s}

takes

20 \text{ s}

to stop with constant deceleration. Find the train's instantaneous speed at

t = 8 \text{ s}

after braking begins.

← Back to Instantaneous Speed Practice Mode

Related Concepts

Velocity Average Speed

Background Knowledge

These ideas may be useful before you work through the harder examples.

velocityaverage speed