Motion in Two and Three Dimensions
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Three-dimensional motion with constant acceleration: |
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When an object moves in such a way that its position can be expressed by three coordinates
(x, y, z) then the object is said to be moving in three dimensions.
In three dimensions the position vector of an object, its displacement, velocity and
acceleration are expressed in terms of three components. However, in two dimensions these
physical quantities are expressed in terms of two components.
Let a = axi + ayj + azk is constant
so the components ax, ay, and az are constant.
Let at t = 0 a particle is at position r0 = x0i + y0j + z0k
and has velocity v0 = v0xi + v0yj + v0zk
after a time interval t, its velocity has changed by an amount
Δv = at. We can rewrite this in terms of
the components as
Δvxi + Δvyj + Δvzk = axti + aytj + aztk.
Comparing the coefficients of i, j & k on both sides we get:
Δvx = axt, Δvy = ayt, Δvz = azt.
We therefore have at time t:
vx = v0x + axt,
vy = v0y + ayt, vz = v0zt + azt,
or in vector form the resultant velocity vector is
v = v0 + at.
Similarly applying second law of motion to get positions in x, y & z
direction
x = x0 + v0xt + (1/2)axt2, y = y0 + v0yt + (1/2) ayt2, z = z0 + v0zt + (1/2)azt2
Newton’s Laws in Three-Dimensional Vector Form:
So the total force exerted on a body is given as the product of mass (m) and its acceleration (a)
F = m (axi + ayj + azk)
Or F= F1i+ F2j + F3 k = maxi + mayj + mazk
so that we get F1 = max, F2 = may, F3 = maz.
This expression shows Newton’s second law in three-dimensional vector form. Newton’s first and
third laws do not have mathematical form and are not required to express in three-dimensional
vector form.
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Newtonian Mechanics