Use
a(t) = −9.8
meters per second per second as the acceleration due to gravity. (Neglect air resistance.)
A canyon is 2300 meters deep at its deepest point. A rock is dropped from the rim above this point. Write the height of the rock as a function of time t in seconds. (Use the canyon floor as position 0.)
f(t) = ?
To find the height of the rock as a function of time, we need to use the formula for position as a function of time under constant acceleration:
f(t) = f(0) + v(0)t + (1/2)at^2
In this case, since the rock is dropped from rest (v(0) = 0), the equation simplifies to:
f(t) = f(0) + (1/2)at^2
Given that the initial position is the rim of the canyon, which is 2300 meters above the canyon floor, we have f(0) = 2300 meters.
Substituting the value of acceleration due to gravity, a(t) = -9.8 meters per second per second, we have:
f(t) = 2300 + (1/2)(-9.8)t^2
Therefore, the height of the rock as a function of time t in seconds is:
f(t) = 2300 - 4.9t^2
To write the height of the rock as a function of time t, we need to first determine the initial conditions. Since the rock is dropped from the rim, we can assume that the initial height of the rock is the depth of the canyon, which is 2300 meters.
Next, we can use the equation of motion to describe the height of the rock as a function of time. The equation of motion for an object in free fall (neglecting air resistance) is given by:
h(t) = h0 + v0t + (1/2)at^2
Where:
h(t) is the height of the object at time t
h0 is the initial height of the object
v0 is the initial velocity of the object
a is the acceleration due to gravity
In this problem, the initial height h0 is 2300 meters, the initial velocity v0 is 0 m/s (since the rock is dropped), and the acceleration due to gravity, a, is -9.8 m/s^2 (negative sign indicates downward direction).
Substituting the values into the equation, we get:
h(t) = 2300 + 0t + (1/2)(-9.8)t^2
Simplifying the equation further, we have:
h(t) = 2300 - 4.9t^2
Therefore, the height of the rock as a function of time t is given by:
f(t) = 2300 - 4.9t^2
a = -9.8
v = Vi - 9.8 t
h = Hi + Vi t +(1/2) a t^2
h = 2300 + 0 t - 4.9 t^2
= 2300 - 4.9 t^2