Give acceleration a=d^2s/dt^2,initial velocity, and initial position of a body moving on a coordinate line. Find the bodys position at time t.

a=e^t v(0)=20 s(0)=5

To find the body's position at time t, we need to integrate the given expression for acceleration, a, twice with respect to time.

Given:
a = e^t
v(0) = 20 (initial velocity)
s(0) = 5 (initial position)

First, we integrate the expression for acceleration, a, to obtain the velocity function, v(t):
v(t) = ∫ a dt
= ∫ e^t dt
= e^t + C1

Next, we use the given initial velocity, v(0) = 20, to determine the value of the constant, C1:
v(0) = e^0 + C1
20 = 1 + C1
C1 = 20 - 1
C1 = 19

Therefore, the velocity function, v(t), is:
v(t) = e^t + 19

Now, we integrate the velocity function, v(t), to obtain the position function, s(t):
s(t) = ∫ v(t) dt
= ∫ (e^t + 19) dt
= e^t + 19t + C2

Next, we use the given initial position, s(0) = 5, to determine the value of the constant, C2:
s(0) = e^0 + 19(0) + C2
5 = 1 + C2
C2 = 5 - 1
C2 = 4

Therefore, the position function, s(t), is:
s(t) = e^t + 19t + 4

Finally, to find the body's position at time t, substitute the given value of t into the position function, s(t):
s(t) = e^t + 19t + 4

Substituting t = 0 into the equation:
s(0) = e^0 + 19(0) + 4
s(0) = 1 + 0 + 4
s(0) = 5

Thus, the body's position at time t is s(t) = e^t + 19t + 4, where t is the desired time value.