Relative Motion in Kinematics 🚗🚶

students, imagine you are on a moving bus and you toss a ball straight up. To you, the ball rises and falls straight back into your hand. To someone standing on the sidewalk, the ball follows a curved path because it keeps moving forward with the bus while also moving up and down. That difference is the heart of relative motion. In Solid Mechanics 1, relative motion helps us describe how objects move compared with a chosen observer or reference frame.

What Relative Motion Means

In kinematics, motion is always measured relative to something. An object may be at rest for one observer and moving for another. Relative motion is the study of how the position, velocity, and acceleration of one body are related to those of another body or a reference frame.

The key idea is simple: motion is not absolute. We must always ask, “Relative to what?” 👀

If body $A$ moves with position vector $\mathbf{r}_A$ and body $B$ moves with position vector $\mathbf{r}_B$, then the position of $A$ relative to $B$ is

$$\mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B$$

This means the relative position vector points from $B$ to $A$.

From this, we can define relative velocity and relative acceleration by differentiating with respect to time:

$$\mathbf{v}_{A/B} = \mathbf{v}_A - \mathbf{v}_B$$

$$\mathbf{a}_{A/B} = \mathbf{a}_A - \mathbf{a}_B$$

These equations are the foundation of relative motion in simple translational motion.

Why Relative Motion Matters in Real Life

Relative motion appears everywhere in engineering and daily life. If you are walking inside a moving train, your speed relative to the train is not the same as your speed relative to the ground. A drone flying in the wind also has one velocity relative to the air and another relative to the Earth 🌍.

For example, suppose a car moves east at $20\,\text{m/s}$ and a passenger walks east inside the car at $2\,\text{m/s}$ relative to the car. The passenger’s velocity relative to the ground is

$$\mathbf{v}_{P/G} = \mathbf{v}_{P/C} + \mathbf{v}_{C/G}$$

So the passenger moves at $22\,\text{m/s}$ east relative to the ground. If the passenger walked west at $2\,\text{m/s}$ inside the car, then the ground speed would be $18\,\text{m/s}$ east.

This kind of reasoning is useful in mechanics because many problems involve two moving bodies, such as:

a piston and a connecting rod,
a vehicle and a road,
a boat and a river current,
an elevator and a person inside it.

Reference Frames and Sign Convention

To solve relative motion problems correctly, students, you need a clear reference frame. A reference frame is the viewpoint from which motion is measured. In many basic problems, one frame is taken as fixed to the ground, and another frame is attached to a moving body.

A few important rules help avoid mistakes:

Choose a positive direction and keep it consistent.
Write all velocities and accelerations with signs.
Separate the motion into components if the motion is in two dimensions.
Check whether the frame itself is moving with constant velocity or accelerating.

For motion along a straight line, if $x_A(t)$ and $x_B(t)$ are the positions of two objects, then the relative position is

$$x_{A/B}(t) = x_A(t) - x_B(t)$$

Differentiating gives

$$v_{A/B}(t) = \frac{dx_{A/B}}{dt} = v_A(t) - v_B(t)$$

and

$$a_{A/B}(t) = \frac{dv_{A/B}}{dt} = a_A(t) - a_B(t)$$

These are especially useful in rectilinear motion, where everything happens along one line.

Relative Motion in Rectilinear Motion

Rectilinear motion means motion along a straight line. Relative motion in this case is the easiest to visualize because we can treat everything as positive or negative along one axis.

Example 1: Two Cars on a Straight Road

Car $A$ travels east at $30\,\text{m/s}$, and car $B$ travels east at $18\,\text{m/s}$. The velocity of $A$ relative to $B$ is

$$v_{A/B} = v_A - v_B = 30 - 18 = 12\,\text{m/s}$$

So $A$ appears to move forward past $B$ at $12\,\text{m/s}$.

If instead car $B$ travels west at $18\,\text{m/s}$, then using east as positive,

$$v_B = -18\,\text{m/s}$$

and

$$v_{A/B} = 30 - (-18) = 48\,\text{m/s}$$

This large relative speed makes sense because the cars are moving toward opposite directions.

Example 2: Overtaking Problem

Suppose a truck travels at $15\,\text{m/s}$ and a car behind it travels at $25\,\text{m/s}$. The closing speed of the car relative to the truck is

$$v_{C/T} = 25 - 15 = 10\,\text{m/s}$$

If the car is initially $50\,\text{m}$ behind the truck, the time needed to catch up is

$$t = \frac{50}{10} = 5\,\text{s}$$

Relative motion turns a harder two-body problem into a simpler one-body problem.

Relative Motion in Curvilinear Motion

Curvilinear motion is motion along a curved path. This is common when objects move in two dimensions, such as a plane in flight or a ball thrown from a moving vehicle.

In two-dimensional relative motion, vectors are used instead of just signed numbers. The same relationship still holds:

$$\mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B$$

$$\mathbf{v}_{A/B} = \mathbf{v}_A - \mathbf{v}_B$$

$$\mathbf{a}_{A/B} = \mathbf{a}_A - \mathbf{a}_B$$

These vector equations can be resolved into components along the $x$- and $y$-axes.

Example 3: River Boat Problem

A boat moves across a river with velocity $\mathbf{v}_{B/W}$ relative to the water, while the water itself moves downstream with velocity $\mathbf{v}_{W/G}$ relative to the ground. The boat’s velocity relative to the ground is

$$\mathbf{v}_{B/G} = \mathbf{v}_{B/W} + \mathbf{v}_{W/G}$$

If the boat points straight across the river and the river current flows downstream, the actual path of the boat relative to the ground is diagonal. This is a classic example of vector addition in relative motion.

Example 4: Plane and Wind

A plane may fly at $200\,\text{m/s}$ relative to the air, while the wind moves east at $30\,\text{m/s}$. If the plane heads north, then its ground velocity has a north component of $200\,\text{m/s}$ and an east component of $30\,\text{m/s}$. Its speed relative to the ground is

$$v_{P/G} = \sqrt{200^2 + 30^2}$$

This gives a slightly larger value than $200\,\text{m/s}$ because the wind adds to the plane’s motion.

Relative Acceleration

Relative acceleration is important when the two bodies are speeding up, slowing down, or changing direction. The general rule is still simple:

$$\mathbf{a}_{A/B} = \mathbf{a}_A - \mathbf{a}_B$$

If a train accelerates forward and a passenger walks inside it, the passenger’s acceleration relative to the ground depends on both the passenger’s motion inside the train and the train’s acceleration.

A useful idea is that if the reference frame has constant velocity, relative acceleration is found by direct subtraction. If the reference frame itself accelerates, the situation becomes more advanced, but the same vector idea remains the starting point.

Example 5: Train Starting from Rest

A train accelerates east at $1.5\,\text{m/s}^2$. A person walks forward inside the train with acceleration $0.5\,\text{m/s}^2$ relative to the train. Then the person’s acceleration relative to the ground is

$$a_{P/G} = 0.5 + 1.5 = 2.0\,\text{m/s}^2$$

if both accelerations point east. If the person walked backward relative to the train, the signs would change accordingly.

How to Solve Relative Motion Problems

students, a reliable method is to follow these steps:

Identify the bodies and the observer or reference frame.
Choose a positive direction or coordinate system.
Write the position, velocity, or acceleration relationships.
Break vector motion into components if needed.
Substitute known values carefully with signs.
Check the result for physical sense.

For example, if one object is moving faster than another in the same direction, the relative speed should be smaller than the larger speed but not negative unless the direction is opposite.

A common mistake is to mix up absolute motion with relative motion. Another is to use scalar speed where vector velocity is needed. In mechanics, direction matters just as much as size ⚙️.

Connection to the Bigger Picture of Kinematics

Relative motion is a central part of kinematics because kinematics describes motion without directly considering forces. Position, velocity, and acceleration are the main quantities, and relative motion shows how these quantities are compared between bodies.

In Solid Mechanics 1, relative motion helps build the foundation for more advanced topics where moving links, mechanisms, and components must be analyzed together. It connects rectilinear motion and curvilinear motion by using the same basic vector relationships in one and two dimensions.

When you understand relative motion, you can interpret motion more clearly in real systems like vehicles, machines, and fluids interacting with structures. This makes it an essential tool for solving engineering problems accurately.

Conclusion

Relative motion is the study of how one object moves as seen from another object or frame. The main equations are

$$\mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B$$

$$\mathbf{v}_{A/B} = \mathbf{v}_A - \mathbf{v}_B$$

$$\mathbf{a}_{A/B} = \mathbf{a}_A - \mathbf{a}_B$$

These relationships work for straight-line motion and for curved motion when written in vector form. By choosing a clear frame, keeping track of signs, and using component analysis, students, you can solve many kinematics problems involving moving bodies with confidence ✅.

Study Notes

Relative motion compares the motion of one body with respect to another body or reference frame.
Always ask, “Relative to what?” before solving a problem.
Relative position is given by $\mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B$.
Relative velocity is given by $\mathbf{v}_{A/B} = \mathbf{v}_A - \mathbf{v}_B$.
Relative acceleration is given by $\mathbf{a}_{A/B} = \mathbf{a}_A - \mathbf{a}_B$.
In rectilinear motion, use signed scalars along one line.
In curvilinear motion, use vectors and resolve them into components.
Common examples include cars on roads, passengers in trains, boats in rivers, and planes in wind.
A moving frame can make an object appear stationary, faster, slower, or moving in a different direction.
Relative motion is a key part of kinematics and supports the analysis of more complex mechanical systems.