Answer
(a) The barrel must be 4.38 meters long.
(b) The speed of the objects will be 300 m/s.
(c) (i) $F = 2.70\times 10^4 ~N$
(ii) $F = 9.00\times 10^3 ~N$
Work Step by Step
(a) $x = (9.00\times 10^3~m/s^2)~t^2 - (8.0\times 10^4~m/s^3)~t^3$
At t = 0.025 s:
$x = (9.00\times 10^3~m/s^2)(0.025~s)^2 - (8.0\times 10^4~m/s^3)(0025~s)^3$
$x = 4.38~m$
The barrel must be 4.38 meters long.
(b) $x = (9.00\times 10^3~m/s^2)~t^2 - (8.0\times 10^4~m/s^3)~t^3$
$v = \frac{dx}{dt} = (1.80\times 10^4~m/s^2)~t - (2.40\times 10^5~m/s^3)~t^2$
At t = 0.025 s:
$v = (1.80\times 10^4~m/s^2)(0.025~s) - (2.40\times 10^5~m/s^3)(0.025)^2$
$v = 300 ~m/s$
The speed of the objects will be 300 m/s.
(c) $v = (1.80\times 10^4~m/s^2)~t - (2.40\times 10^5~m/s^3)~t^2$
$a = \frac{dv}{dt} = (1.80\times 10^4~m/s^2) - (4.80\times 10^5~m/s^3)~t$
(i) At t = 0:
$a = \frac{dv}{dt} = (1.80\times 10^4~m/s^2) - (4.80\times 10^5~m/s^3)~t$
$a = (1.80\times 10^4)~m/s^2$
$F = ma = (1.50~kg)((1.80\times 10^4)~m/s^2)$
$F = 2.70\times 10^4 ~N$
(ii) At t = 0.025 s:
$a = (1.80\times 10^4~m/s^2) - (4.80\times 10^5~m/s^3)(0.025~s)$
$a = (6.00\times 10^3)~m/s^2$
$F = ma = (1.50~kg)((6.00\times 10^3)~m/s^2)$
$F = 9.00\times 10^3 ~N$
