1-D Motion with Constant Acceleration Calculator

Calculate the final velocity and displacement for an object moving with constant acceleration using:
\[ v = u + at \quad \text{and} \quad s = ut + \frac{1}{2}at^2. \]

* Enter the initial velocity \( u \) (m/s), acceleration \( a \) (m/s²), and time \( t \) (s) in SI units.

Step 1: Enter Parameters

Example: 0 m/s

Example: 9.81 m/s²

Example: 5 s

Equations used:
\( v = u + at \)
\( s = ut + \frac{1}{2}at^2 \)


Practical Example:
For an object starting from rest (\( u = 0 \) m/s) with an acceleration of 9.81 m/s² over 5 s:
Final velocity: \( v = 0 + 9.81 \times 5 \approx 49.05\, \text{m/s} \)
Displacement: \( s = 0 \times 5 + \frac{1}{2} \times 9.81 \times 5^2 \approx 122.63\, \text{m} \)

1D Motion with Constant Acceleration

1D Motion with Constant Acceleration

One-dimensional (1D) motion with constant acceleration is a fundamental topic in kinematics. It helps describe the motion of an object moving along a straight line when the rate of change of its velocity (i.e., the acceleration) is constant.

Key Equations of Motion

The standard equations for 1D motion with constant acceleration (a) assume:

  • x is the object’s position.
  • v is the object’s velocity.
  • t is the time.
  • x0 and v0 are the initial position and velocity, respectively.
  • a is the constant acceleration.

We often use three core equations:

1) v = v0 + a t
2) x = x0 + v0 t + (1/2) a t2
3) v2 = v02 + 2 a (x − x0)

These equations allow you to determine the position and velocity of an object at any time t, given an initial state (x0, v0) and a constant acceleration a.

Detailed Breakdown

1) Velocity as a Function of Time

v = v0 + a t

This equation tells us how the velocity changes over time. Starting from an initial velocity v0, after time t, the new velocity v is:

  • The initial velocity v0
  • Plus the product of acceleration a and elapsed time t.

2) Position as a Function of Time

x = x0 + v0 t + (1/2) a t2

This equation tells us the position x after time t when the acceleration a is constant. It consists of three parts:

  • x0: the initial position
  • v0 t: the displacement from the initial velocity over time
  • (1/2) a t2: the displacement due to acceleration

3) Velocity as a Function of Displacement

v2 = v02 + 2 a (x − x0)

This form is handy when you need to relate velocity directly to displacement without involving time explicitly. It shows how changes in position translate into changes in velocity for constant acceleration.

Example 1: Simple Velocity Update

Suppose an object starts at rest (v0 = 0) and undergoes a constant acceleration of a = 2 m/s2. Find its velocity after 5 seconds.

v = v0 + a t = 0 + (2 m/s2) × (5 s) = 10 m/s

So, after 5 seconds, the object’s velocity is 10 m/s.

Example 2: Position Update

Let’s use the same scenario as above but find the position after 5 seconds. Assume the object starts from x0 = 0.

x = x0 + v0 t + (1/2) a t2

Substituting the values:

x = 0 + (0 × 5) + (1/2) × (2 m/s2) × (5 s)2
x = 0 + 0 + 1 × 25
x = 25 m

After 5 seconds, the object has traveled 25 meters.

Example 3: Velocity from Displacement

Suppose the same object travels from x0 = 0 to x = 25 m under constant acceleration a = 2 m/s2. What is the velocity when it reaches 25 m?

v2 = v02 + 2 a (x − x0)

Since v0 = 0 and x − x0 = 25 m:

v2 = 0 + 2 × (2 m/s2) × 25 m
v2 = 100 (m2/s2)
v = 10 m/s

So, the velocity at 25 m is 10 m/s, matching our previous calculation.

Key Takeaways:
  • Use v = v0 + a t to find velocity after time t.
  • Use x = x0 + v0 t + (1/2) a t2 to find position after time t.
  • Use v2 = v02 + 2 a (x − x0) to find velocity from displacement.
  • All these equations are valid only under constant acceleration.

With these equations, you can analyze a wide range of everyday and engineering scenarios involving constant acceleration, from free-falling objects under gravity to vehicles accelerating along a straight road.