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 7.56 m/s. (a) What is the acceleration (magnitude and direction) of the pebble during the downward motion? (b) After 0.950 s, how far beneath the cliff top is the pebble?
a. -9.8m/s^2
b. d=vi*t+ 1/2 at^2=-7.56*t-4.96t^2
To answer these questions, we need to consider the acceleration due to gravity, which acts in the downward direction. We can use the equations of motion to find the answers.
(a) Acceleration of the pebble during downward motion:
The acceleration due to gravity near the Earth's surface is approximately 9.8 m/s². Since the pebble is falling downward, the acceleration will be in the negative direction.
Therefore, the magnitude of the acceleration is 9.8 m/s², and the direction is downward.
(b) Distance beneath the cliff top after 0.950 s:
We can use the kinematic equation to determine the distance:
d = v₀t + (1/2)at²
where:
d = distance
v₀ = initial velocity
t = time
a = acceleration
Plugging in the given values:
v₀ = 7.56 m/s
t = 0.950 s
a = -9.8 m/s² (negative because it is in the downward direction)
d = (7.56 m/s)(0.950 s) + (1/2)(-9.8 m/s²)(0.950 s)²
Simplifying the equation:
d = 7.182 m - 4.419 m
d = 2.763 m (rounded to three decimal places)
Therefore, after 0.950 s, the pebble is approximately 2.763 meters beneath the cliff top.
To find the acceleration of the pebble during the downward motion, we need to use the kinematic equation:
v = u + at
Where:
- v is the final velocity (which is zero since the pebble is launched straight downward),
- u is the initial velocity (7.56 m/s downwards),
- a is the acceleration, and
- t is the time taken.
To find the acceleration (magnitude and direction), we need to rearrange the equation to solve for a:
a = (v - u) / t
Since the final velocity (v) is zero, the equation becomes:
a = -u / t
Substituting the given values:
u = -7.56 m/s
t = t = 0.950 s
a = (-(-7.56 m/s)) / 0.950 s = 7.56 m/s / 0.950 s = 7.9684 m/s²
The magnitude of the acceleration is 7.9684 m/s², and since the pebble is fired downward, the direction of acceleration is also downward (negative).
Now, to find how far beneath the cliff top the pebble is after 0.950 s, we can use another kinematic equation:
s = ut + (1/2)at²
Where:
- s is the displacement,
- u is the initial velocity,
- t is the time, and
- a is the acceleration.
Since the pebble is initially fired downward, the displacement will be negative.
Substituting the given values:
u = -7.56 m/s (downward)
t = 0.950 s
a = -7.9684 m/s² (downward)
s = (-7.56 m/s)(0.950 s) + (1/2)(-7.9684 m/s²)(0.950 s)²
s = -7.182 m - 3.41586 m
s = -10.59786 m
Therefore, after 0.950 s, the pebble is approximately 10.6 m beneath the cliff top.