Scalars and Vectors：練習問題

A car is traveling along a straight road at a constant speed of \( 20 \text{ m/s} \). The driver applies the brakes, providing a constant deceleration of \( 4 \text{ m/s}^2 \). What is the total displacement of the car from the moment the brakes are applied until it comes to a complete stop?

A pilot points an aircraft due North with an airspeed of \( 200\text{ km/h} \). A wind is blowing from the West at a constant speed of \( 50\text{ km/h} \). To maintain a track that is exactly due North, at what angle relative to the North-South line should the pilot point the aircraft, and what will be the resulting ground speed?

A car accelerates from rest along a straight road. The acceleration is given by \( a(t) = \alpha t - \beta t^2 \), where \( \alpha = 6\text{ m/s}^3 \) and \( \beta = 1.2\text{ m/s}^4 \). What is the maximum velocity attained by the car before it begins to slow down?

The speed of a particle moving along a straight path is given by the expression \( v(s) = k s^2 \), where \( s \) is the displacement from the origin and \( k \) is a positive constant. Which of the following expressions represents the acceleration \( a \) of the particle as a function of \( s \)?

A stone is dropped from a height \( H \). Simultaneously, another stone is thrown upward from the ground with an initial velocity \( v_0 \). The two stones collide at a height of \( \frac{H}{3} \). Express \( v_0 \) in terms of \( g \) and \( H \).

An object starts from rest and moves along a straight line with an acceleration given by \( a(t) = kt^{1/2} \), where \( k \) is a constant. What is the ratio of the object's displacement at time \( t = T \) to its displacement at time \( t = 4T \)?

A particle moves in the \( xy \)-plane such that its position vector \( \vec{r}(t) \) and velocity vector \( \vec{v}(t) \) satisfy the condition \( \vec{r} \cdot \vec{v} = 0 \) at all times. Show that the distance of the particle from the origin remains constant and identify the shape of its trajectory.

A particle of mass \( m \) moves in a straight line through a fluid that exerts a resistive force proportional to the velocity raised to the power of \( 3/2 \), such that the acceleration is given by \( a = -k v^{3/2} \), where \( k \) is a positive constant. The particle has an initial velocity \( v_0 \) at time \( t = 0 \) and position \( x = 0 \).

(a) Derive an expression for the velocity \( v(t) \) as a function of time.
(b) Determine the time \( t_s \) required for the particle to slow down to one-quarter of its initial velocity.
(c) Find the total distance \( D \) the particle travels as \( t \to \infty \). Explain the physical significance of your result.