Answer:
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.
Step-by-step explanation:
I´ve found the complete question on the web:
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.