Linear motion under a variable force：練習問題

A particle of mass \(m\) moves away from the origin \(O\) along the \(x\)-axis. It is acted upon by a force \(F = \frac{m}{x+1}\) in the direction of increasing \(x\). If the particle starts from rest at \(x=0\), find the speed of the particle when it reaches \(x = e^2 - 1\).

A particle of mass \(m\) moves along a straight line. It is acted upon by a force \(F = P - kv\) in the direction of motion, where \(v\) is the velocity and \(P, k\) are positive constants. The particle starts from rest at \(t=0\). Find the time \(t\) taken for the particle to reach a velocity of \(v = \frac{P}{2k}\).

A particle of mass \(m\) moves in a straight line under a resistive force of magnitude \(mkv^2\), where \(v\) is its velocity and \(k\) is a positive constant. At time \(t = 0\), the particle has velocity \(u\). Find the time taken for the velocity of the particle to reduce to \(\frac{1}{3}u\).

A particle of mass \(m\) moves in a straight line under a force \(F = \frac{k}{x^2}\) directed away from the origin \(O\), where \(x\) is the distance from \(O\). The particle starts from rest at \(x = a\). Find the speed of the particle when it is at a distance \(2a\) from the origin.

A particle moves in a straight line such that its acceleration at distance \(x\) from a fixed point \(O\) is given by \(a = \frac{1}{x^3}\). If the particle starts from rest at \(x = 2\), find the speed of the particle when it reaches \(x = 4\).

A particle of mass \(m\) moves along a straight line under the action of a force \(F = m(k^2 + v^2)\), where \(v\) is the velocity and \(k\) is a constant. If the particle starts from rest at \(x = 0\), find an expression for the distance \(x\) in terms of \(v\) and \(k\).

A particle of mass \( m \) moves in a straight line on a horizontal surface. It is acted upon by a resistive force of magnitude \( m(kv + cv^2) \), where \( v \) is its speed and \( k, c \) are positive constants. The particle is projected with an initial speed \( U \) at time \( t = 0 \).

(a) Show that the time \( t \) taken for the speed to decrease to \( v \) is given by \( t = \frac{1}{k} \ln\left( \frac{U(k + cv)}{v(k + cU)} \right) \).
(b) Find an expression for the distance \( x \) traveled by the particle in terms of \( v \) and the given constants, assuming \( x=0 \) when \( t=0 \).