A particle experiences a constant acceleration that is south at 2.50 meters per second squared. At t=0, its velocity is 40.0 meters per second east. What is its velocity at t=8.00 seconds?
The acceleration is due south, so the east-west component (40 m/s) is not altered.
For the north-south component, at t=0,
initial velocity = 0 m/s
after 8 seconds,
velocity v=0+(-2.5)8 = -20 m/s (<0 for south)
Velocity = (40,-20) m/s
Magnitude = √(40²+20²)
=44.72 m/s
direction:
atan(-20/40)=-26.57°, or
26.57° south of east.
To find the velocity of the particle at t=8.00 seconds, we need to use the equations of motion under constant acceleration.
The equation we will use is:
v = u + at
Where:
v = final velocity
u = initial velocity
a = acceleration
t = time
In this problem:
u = 40.0 meters per second east (initial velocity)
a = -2.50 meters per second squared south (acceleration)
t = 8.00 seconds (time)
First, let's determine the direction of the final velocity. Since the acceleration is south, it will reduce the magnitude of the velocity in the east direction. Therefore, the final velocity will have both south and east components.
To calculate the final velocity, we'll use the equation in vector form:
v = v_xi + v_yj
Where:
v = final velocity vector
v_x = final velocity in the x-direction (east)
v_y = final velocity in the y-direction (south)
To find v_x, we use the equation:
v_x = u_x + a_xt
Where:
u_x = initial velocity in the x-direction (east)
a_x = acceleration in the x-direction (east)
t = time
Since there is no acceleration in the east direction, a_x = 0. Therefore, the x-component of velocity remains constant throughout the motion.
Substituting the given values:
v_x = u_x + a_xt
v_x = 40.0 meters per second east + 0 * 8.00 seconds
v_x = 40.0 meters per second east
Now, let's calculate v_y, the vertical component of the final velocity.
v_y = u_y + a_yt
Where:
u_y = initial velocity in the y-direction (south)
a_y = acceleration in the y-direction (south)
t = time
Substituting the given values:
v_y = u_y + a_yt
v_y = 0 meters per second south + (-2.50 meters per second squared south) * 8.00 seconds
v_y = -20.0 meters per second south
Now that we have both components, we can write the final velocity vector:
v = v_xi + v_yj
v = (40.0 meters per second east) + (-20.0 meters per second south)
The final velocity is 40.0 meters per second east - 20.0 meters per second south.