Question:
A mysterious rocket-propelled object of mass.. A mysterious rocket-propelled object of mass 45.0 kg is initially at rest in the middle of the horizontal, frictionless surface of an ice-covered lake. Then a force directed east and with magnitude F(t) = (16.8N/s)t is applied. How far does the object travel in the first 5.00 s after the force is applied?
Answer:
The object travels 7.53 m in the first 5.00 s after the force is applied
Explanation:
Let the force F be given by
F = ma
Where
m = Mass = 46.5 kg
a = Acceleration
We can express the force as a function of time as follows
F = pt
p = variable of F as a function of time
t = Time
Therefore a = pt/m
∴ [tex]\frac{dv}{dt} = \frac{pt}{m}[/tex]
[tex]{dv} = \frac{pt}{m}dt[/tex]
[tex]{\int\limits^v_0 dv} =\int\limits^t_0 \frac{pt}{m}dt[/tex]
v = [tex]\frac{pt^2}{2m}[/tex]
[tex]\frac{dx}{dt}[/tex]=[tex]\frac{pt^2}{2m}[/tex]
[tex]dx = \frac{pt^2}{2m} dt[/tex]
[tex]\int\limits^x_0 \, dx = \int\limits^t_0 \frac{pt^2}{2m} dt[/tex]
x = [tex]\frac{pt^3}{6m}[/tex]
x = Distance
After 5 seconds F = 16.8*5 = 84 N and
a = F/a = 1.8 m/s² and
p = ma/t = 16.8 N/s
x = 16.8*5³÷(6*46.5)
= 7.53 m
