May 2, 1997

Physics 211, Astronomy 211 - A car is moving with constant velocity so that its position coordinate x at time t is equal to x = at + b, where a and b are constants. Solve this equation for t. (Algebra.)

May 9, 1997

Physics 211, Astronomy 211 - This question is concerned with the graph of the equation x = at + b, from the May 2, 1997 question. Graphs are used a great deal in physics and other sciences because they allow you to collect a lot of information in a form that can easily be visualized. (Graphs. Algebra.)

(a)The graph of this equation is a straight line (as long as the t and x axes use linear scales, i.e., scales on which equal increments of t and x are represented by equal distances). Why is this so? What kind of an equation relating x to t would result in a nonlinear graph?

(b)Which quantities on the graph represent the quantities a and b? Illustrate this by drawing the graphs corresponding to (i) a = 1.5 and b = -2 and (ii) a = -0.5 and b = 1. Put t on the horizontal axis and x on the vertical axis.

(c)For the two graphs in part (b), determine the intercepts with the t-axis. Do the calculated values agree with those shown by the graphs? What is the physical interpretation of these intercepts?

(d)Now graph the two cases (i) and (ii) from part (b) so that x is on the horizontal axis and t is on the vertical axis. What are the physical interpretations of the intercepts with the two axes now? Do the values of the intercepts as shown by the graphs agree with the values that can be calculated from the given equation?

(e)Determine the slopes of the graphs of x vs. t and t vx. x obtained in parts (b) and (d), respectively. How are these slopes related? Can you explain why the slopes are related the way they are?

May 16, 1997

Physics 211, Astronomy 211 - A car is moving with constant speed along a straight line. (This is the same as saying the car is moving with constant velocity.) Let the straight line be called the x-axis. Suppose that at t = -4 s the car is at x = 100 m and at t = 2 s the car is at x=-20 m. Plot an x vs. t graph of the car's motion, and derive an equation relating the car's position x to the time t. (Analytical geometry.)

Hint: Velocity is defined as (change in position)/(change in time). Here, the velocity is given to be constant.

May 23, 1997

Physics 211, Astronomy 211 - This question is related to the May 16 question. Suppose a second car is moving along the same straight line such that its position x at time t is given by the equation x = 10t + 60, where t is measured in seconds and x in meters. At what time and at what position do the two cars pass each other? (Algebra. Graphs.)

Solve this problem both analytically, i.e., by means of a calculation, and graphically.

May 30, 1997

Physics 211 - (This is a challenge question. It contains concepts that you are not expected to have studied before beginning the course.) Let us consider once more the car from the May 16 question. The position x of the car as a function of the time t could be described by the equation x = -20t + 20. (Analytical Geometry. Graphs.)

Suppose you have a reference frame, called the car's frame, in which this car is at rest, i.e., a reference frame that is moving at the car's velocity relative to the road. The frame in which the road is at rest is called the road's frame. In the car's frame, the car's position is constant. Let x' denote the position coordinate in the car's frame. (x is the position coordinate in the road's frame.) Let us choose the origin of the car's frame, x' = 0, so that the constant position of the car in this frame is equal to x' = 20 m, at all times.

Hint: You can think of the car's frame as an (infinitely) long ruler rigidly attached to the car, aligned with the direction in which the car is moving, and moving with the car. The ruler is used for measuring the positions of objects. Thus, to determine the value of the position of an object in the car's frame at a given time, i.e., the object's x'-value at that time, one simply records where on the ruler the object is at that time. The car itself is at x' = 20 m on this ruler. The origin of the ruler is outside of the car.

(a).Using the same set of x,t axes as for the x vs. t graph of the motion of the car in the May 16 question, show the x vs. t graph of a second car that is at rest at the origin of the car's frame, i.e., for which x' = 0 at all times.

(b).Do the same as in part (a), but for a car that is at rest at x' = 40 m.

(c).What are the distances between the three cars (i) in the frame in which the road is at rest and (ii) in the car's frame?