Question

A car drives 84 meters forward. It's displacement and distance would be the same?​

Answer 1

Answer:

Yes.

Explanation:

Displacement is a vector quantity, meaning it has to be a straight graph jointing from starting point to final point and has a direction as well as magnitude and it can also be negative as well as positive and zero.

Displacement is also changes in position (points) of distance. A distance is scalar quantity meaning it only has magnitude and does not have direction. Since distance is scalar quantity, it cannot be negative. Also graph of distance can be a curve graph or any graph.

So when does displacement equal to distance? If an object is moving in straight path or line in one/fixed direction, both displacement and distance are same.

Because if a distance is a straight path and the displacement is simply a straight line (or ray) jointing from starting point to end point, both distance and displacement are both straight path or rays. Since distance and displacement have same graph, we can conclude that both are same in values.

If we go by calculus, given x = 2t as example of straight graph where x stands for position and t stands for time. We can find the displacement from b to a by using the following formulas:

Displacement

$\displaystyle \large{s=\int\limits^b_a {v} \, dt}$

Distance

$\displaystyle \large{l=\int\limits^b_a {|v|} \, dt}$

v stands for velocity which we can find from:

Velocity

$\displaystyle \large{v=\frac{dx}{dt}}$

If you have learnt calculus, first, differentiate x = 2t with respect to t.

$\displaystyle \large{v=2}$

Then substitute v = 2 in s and l to find find if both displacement and distance are equal in straight path.

Displacement

$\displaystyle \large{s=\int\limits^b_a {2} \, dt}\\\displaystyle \large{s=[2t]\limits^b_a}\\\displaystyle \large{s=2b-2a = 2(b-a)}$

Distance

$\displaystyle \large{l=\int\limits^b_a {|2|} \, dt}\\\displaystyle \large{l=\int\limits^b_a {2} \, dt}\\\displaystyle \large{l=[2t]\limits^b_a}\\\displaystyle \large{l=2b-2a=2(b-a)}$

Since both displacement and distance are equal when integrating on straight graph, we can conclude that an object moving in straight fixed point has same distance and displacement.