After a boulder is pushed down a hill, its speed s(t) at time t is given by the formula s(t)=v+at, where v is the initial velocity and a is the acceleration. If s(2)=16 and s(5)=25, find the initial velocity and acceleration.(round to the nearest tenth)
To solve this problem, we need to use the given information to create a system of equations and solve for the initial velocity (v) and acceleration (a).
First, let's plug in the values we have into the given formula:
s(2) = v + 2a = 16 (Equation 1)
s(5) = v + 5a = 25 (Equation 2)
Now we have a system of two equations with two variables. We can solve this system to find the values of v and a.
To eliminate v, we can subtract Equation 1 from Equation 2:
(v + 5a) - (v + 2a) = 25 - 16
3a = 9
Divide both sides of the equation by 3 to solve for 'a':
a = 3
Now we can substitute the value of 'a' back into Equation 1 or Equation 2 to solve for 'v'. Let's use Equation 1:
v + 2a = 16
v + 2(3) = 16
v + 6 = 16
v = 16 - 6
v = 10
Therefore, the initial velocity (v) is 10 and the acceleration (a) is 3 (rounded to the nearest tenth).
acceleration: a
v(t) = at + c
it says it was pushed, so we may assume initial velocity is zero, thus
0 = a(0) + c ---> c=0
v(t) = at
s(t) = (1/2)at^2 + k
given: when t=2, s(2) = 16
(1/2)a(4) + k = 16
2a + k = 16 ****
when t=5, s(5) = 25
(1/2)a(25) + k = 25
(25/2)a + k = 25 ***
subtract equations marked ***
(21/2)a = 9
a = 18/21 = 6/7
in 2a + k = 16
12/7 + k = 16
k = 100/7
initial velocity is 0
a = 6/7