题目

Problem

Two particles, $P$ and $Q$ , have masses $3m$ and $2m$ respectively. The particles are connected by a light inextensible string. Initially $P$ and $Q$ are at rest on a smooth horizontal plane with the string slack.

Particle $P$ is then projected along the plane directly away from $Q$ with speed $4u$ . At the same instant, particle $Q$ is projected along the plane in the opposite direction with speed $3u$ .

Find

(a) the common speed of the particles immediately after the string becomes taut,

(3)

(b) the magnitude of the impulse exerted on $Q$ at the instant when the string becomes taut.

(3)

题目中文翻译

两个质点 $P$ 和 $Q$ 的质量分别为 $3m$ 和 $2m$ ，由一根轻且不可伸长的细绳连接。起初， $P$ 和 $Q$ 静止在光滑水平面上，细绳松弛。

随后将质点 $P$ 沿水平面以速率 $4u$ 向直接远离 $Q$ 的方向抛出。同时，将质点 $Q$ 沿水平面以速率 $3u$ 向相反方向抛出。

求

(a) 细绳刚绷紧后两个质点的共同速率，

(3)

(b) 细绳绷紧瞬间 $Q$ 所受冲量的大小。

(3)

解答

(a)

解法一

思路

展开

绳绷紧瞬间的张力冲量是系统内力，水平方向总动量守恒。绳不可伸长，所以绷紧后两粒子沿同一方向以共同速度运动。

答题过程

展开

Take the original direction of $P$ as positive. Immediately before the string becomes taut, the velocities are $4u$ and $-3u$ . Let their common velocity immediately afterwards be $v$ .

Conservation of momentum gives

3m(4u)+2m(-3u)=(3m+2m)v.

Thus

6mu=5mv,\qquad v=\frac65u>0.

The common speed is $\boxed{6u/5}$ , in the original direction of motion of $P$ .

(b)

解法一

思路

展开

对 $Q$ 使用冲量—动量关系，末速度减初速度时必须保留原来反向运动的负号。

答题过程

展开

The impulse on $Q$ is

I_Q=2m\big(\tfrac65u-(-3u)\big) =2m\big(\tfrac{21}{5}u\big) =\boxed{\frac{42mu}{5}}.

This is positive in the chosen direction, so its magnitude is $42mu/5$ .

解法二

思路

展开

轻绳对两粒子的冲量等大反向，也可以从 $P$ 的动量变化求所需大小。

答题过程

展开

The impulse on $P$ is

I_P=3m\big(\tfrac65u-4u\big)=-\frac{42mu}{5}.

The impulse on $Q$ is opposite to this, and therefore has magnitude $\boxed{42mu/5}$ .

M1 2019 Oct Q1

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

解法二

思路

答题过程