A stone is thrown vertically upward from the top of a 30m high building with a velocity of 15m/s. Taking the acceleration of stone as 9.81 m/s2. And taking that as constant, determine a) the velocity v and elevations sy of stone above the ground at any time t b) the maximum altitude reached by the stone c) time when the stone strikes the ground
The kinematics equations for the stone, ignoring air resistance, is as follows:
vi=initial velocity = 15 (upwards)
g=acceleration due to gravity=(-9.81)
xi=initial position = 20 m (above ground)
t=time in seconds from throwing stone upwards.
(a)
x(t)=vi*t+(1/2)gt²
(remember that g is negative)
(b) equate x'(t)=0 and solve for t.
(c) solve for x(t)=0, retain positive root only.
Answer
To determine the velocity and elevation of the stone at any time, we can use the equations of motion for constant acceleration.
Let's break down the problem into parts:
a) The velocity v and elevation sᵧ of the stone above the ground at any time t:
We can use the following equations of motion:
v = u + at -- (Equation 1)
s = ut + 0.5at^2 -- (Equation 2)
Given:
Initial velocity (u) = 15 m/s (upwards)
Acceleration (a) = -9.81 m/s^2 (negative due to the force of gravity)
Initial elevation (s₀) = 30 m (height of the building)
Substituting the values into Equation 1:
v = 15 - 9.81t -- (Equation 3)
Substituting the values into Equation 2:
s = 30 + 15t - 0.5(9.81)t^2 -- (Equation 4)
Now, we have equations to calculate the velocity (v) and elevation (s) of the stone at any time (t).
b) The maximum altitude reached by the stone:
The maximum altitude occurs when the velocity (v) of the stone becomes zero. To find this point, we can set Equation 3 equal to zero and solve for time (t).
0 = 15 - 9.81t
9.81t = 15
t = 15 / 9.81
Substitute the value of t into Equation 4 to find the maximum altitude (s_max).
s_max = 30 + 15(15/9.81) - 0.5(9.81)(15/9.81)^2
Using these equations, you can calculate the maximum altitude reached by the stone.
c) The time when the stone strikes the ground:
To find the time when the stone strikes the ground, we need to solve Equation 4 for s = 0. This will give us the time it takes for the stone to reach the ground.
0 = 30 + 15t - 0.5(9.81)t^2
Rearranging the equation and solving for t using the quadratic formula will give us the desired result.
These calculations will help you determine the velocity, elevation, maximum altitude, and time when the stone strikes the ground.