Aparticle starts from the origin at t = 0 with a velocity of and moves in the xy plane with a constant acceleration of . at the instant the particle's x coordinate is 29 m, what are (a) its y coordinate and (b) its speed?

a) When the x-component of the position of the particle is 29 m, the y-component is 45 m.

b)When the x-component of the position of the particle is 29 m, its speed is 21.8 m/s.

Explanation:

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A particle starts from the origin at t=0 with a velocity of 8.0 j m/s and moves in the x-y plane with a constant acceleration of (4i + 2j) m/s^2. At the instant the particle's x-coordinate is 29m. What are its y-coordinate and speed?

a) First, let´s find at which time the particle is located at 29 m along the x-axis. For that let´s use the equation of position of an object moving in a straight line at constant acceleration:

x = x0 + v0x · t + 1/2 · ax · t²

Where:

x = x-component of the position of the particle at time t

x0 = initial x-position.

v0x = initial x-component of the velocity.

t = time

ax = x-component of the acceleration.

We have the following data:

x = 29 m

x0 = 0 (because the particle starts from the origin, x = 0 and y = 0).

v0x = 0 (the initial velocity only has an y-component).

ax = 4 m/s²

Then, the equation of position gets reduced to this:

x = 1/2 · a · t²

29 m = 1/2 · 4 m/s² · t²

29 m/ 2 m/s² = t²

t = 3.8 s

Now, we can find the y-component of the position of the particle at that time:

y = y0 + v0y · t + 1/2 · ay · t²

Where:

y = y-component of the position of the particle at time t.

y0 = initial y-component of the position.

v0y = initial y-component of the velocity.

ay = y-component of the acceleration

t = time.

We have the following data:

y0 = 0

v0y = 8.0 m/s

ay = 2 m/s²

t = 3.8 s (calculated above)

Then, we can calculate "y" at t = 3.8 s

y = 0 + 8.0 m/s · 3.8 s + 1/2 · 2 m/s² · (3.8 s)²

y = 45 m

When the x-component of the position of the particle is 29 m, the y-component is 45 m.

b) To find the speed of the particle, let´s use the equation of velocity.

For the x-component of the velocity (vx):

vx = v0x + ax · t   (v0x = 0)

vx = 4 m/s² · 3.8 s

vx = 15.2 m/s

The y-component of the velocity will be:

vy = v0y + ay · t

vy = 8.0 m/s + 2 m/s² · 3.8 s

vy = 15.6 m/s

Then, the vector velocity will be:

v = (15.2, 15.6) m/s

To calculate the speed, we have to find the magnitude of the velocity vector:

When the x-component of the position of the particle is 29 m, its speed is 21.8 m/s.

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