From the top of a cliff, a person uses a slingshot to fire a pebble straight downward, which is the negative direction. The initial speed of the pebble is 9.0 m/s.

(a) What is the acceleration (magnitude and direction) of the pebble during the downward motion?

*Is the pebble decelerating?
* After 0.46 s, how far beneath the cliff top is the pebble?

Well, well, well, we've got ourselves a little slingshot adventure here! Let's break it down, shall we?

(a) The acceleration of the pebble during the downward motion is, hold your breath, a whopping 9.8 m/s² in the downward direction. Gravity has got its fingerprints all over this one, my friend.

Is the pebble decelerating, you ask? Oh, no, no, no! It is not decelerating. In fact, it's actually accelerating. Down, down, down it goes!

Now, let's move on to the second part of your question. After 0.46 s, how far beneath the cliff top is the pebble? Well, we can calculate that using good ol' kinematics.

The distance traveled by the pebble can be determined using the formula:

d = v₀t + ½at²

Plugging in the given values:
v₀ = 9.0 m/s (the initial speed)
t = 0.46 s (the time)
and a = 9.8 m/s² (the acceleration),

We get:

d = (9.0 m/s)(0.46 s) + ½(9.8 m/s²)(0.46 s)²

Just crunch those numbers, and you'll find out how far beneath the cliff top our lovely pebble is. Remember to take the negative sign into account since it's moving downward.

To determine the acceleration of the pebble during its downward motion, we first need to understand that the acceleration due to gravity acts in the negative direction.

Using the equation of motion for constant acceleration:

d = vi * t + (1/2) * a * t^2

where:
d = displacement
vi = initial velocity
t = time
a = acceleration

(a) What is the acceleration (magnitude and direction) of the pebble during the downward motion?

Given:
Initial velocity (vi) = 9.0 m/s
Time (t) = 0.46 s

Since the pebble is moving downward, the acceleration due to gravity is acting in the direction opposite to its motion. Hence, the acceleration is -9.8 m/s^2 (magnitude) in the negative direction (direction).

*Is the pebble decelerating?

Since the acceleration due to gravity is acting opposite to the motion of the pebble, the pebble is decelerating.

* After 0.46 s, how far beneath the cliff top is the pebble?

To find the displacement (distance beneath the cliff top), we can use the same equation of motion:

d = vi * t + (1/2) * a * t^2

Substituting the given values:

d = (9.0 m/s) * (0.46 s) + (1/2) * (-9.8 m/s^2) * (0.46 s)^2

Simplifying the equation:

d = 4.14 m - 0.48 m

d = 3.66 m

After 0.46 seconds, the pebble is approximately 3.66 meters beneath the cliff top.

To find the magnitude and direction of the acceleration during the downward motion of the pebble, we need to understand that acceleration is the rate of change of velocity. Since the pebble is moving downward, its velocity is negative. The initial speed of the pebble is given as 9.0 m/s downwards.

(a) magnitude of acceleration:
Acceleration can be calculated using the formula: acceleration = change in velocity / time. However, in this case, as the pebble is moving only in one direction (downwards), we only need to consider the magnitude of the change in velocity.

Since the initial velocity is 9.0 m/s downwards, the change in velocity is equal to the initial velocity. Therefore, the magnitude of acceleration is 9.0 m/s^2.

Direction of acceleration:
Since the pebble is moving downwards, the direction of acceleration will be in the same direction as the velocity. In this case, the acceleration is negative, indicating it is acting in the opposite direction to the motion of the pebble.

*Is the pebble decelerating?
Yes, the pebble is decelerating because its velocity decreases during the downward motion.

To find how far beneath the cliff top the pebble is after 0.46 s, we can use the equation of motion:

displacement = initial velocity * time + (1/2) * acceleration * time^2

Initial velocity = -9.0 m/s (downward)
Time = 0.46 s
Acceleration = -9.0 m/s^2 (downward)

Therefore, substituting these values into the equation, we have:
displacement = -9.0 * 0.46 + (1/2) * (-9.0) * (0.46)^2

Solving this equation gives us:
displacement = -4.14 - 0.94 = -5.08 m

So, after 0.46 s, the pebble is approximately 5.08 meters beneath the cliff top.