An electron’s position is given by , with in seconds and in meters. (a)In unit-vector notation, what is the electron’s velocity? At , what is (b) in unit-vector notation and as (c) a magnitude and (d) an angle relative to the positive direction of the axis?
The position vector of electron,
The velocity is the rate of change of position of an object with respect to time. It is a vector quantity, which has magnitude as well as direction.
The expression for velocity in general can be written as:
Here, is the position vector.
The magnitude of is given as:
Here, and are the x and y components of velocity.
The direction of is given as:
Using equation (i), the velocity of electron is calculated as:
Thus, the velocity of electron is .
Substitute in the above expression to find the velocity.
Thus, the velocity of electron at t=2.0 s is .
Using equation (ii), the magnitude of velocity is calculated as:
Thus, the magnitude of velocity of electron is 16.3 m/s .
Using equation (iii), the direction of electron’s velocity is calculated as:
Thus, the angle of velocity relative to the positive direction of x-axis is .
A trebuchet was a hurling machine built to attack the walls of a castle under siege. A large stone could be hurled against a wall to break apart the wall. The machine was not placed near the wall because then arrows could reach it from the castle wall. Instead, it was positioned so that the stone hit the wall during the second half of its flight. Suppose a stone is launched with a speed of and at an angle of. What is the speed of the stone if it hits the wall (a) just as it reaches the top of its parabolic path and (b) when it has descended to half that height? (c)As a percentage, how much faster is it moving in part (b) than in part (a)?
During a tennis match, a player serves the ball at, with the center of the ball leaving the racquet horizontally above the court surface. The net is away and high. When the ball reaches the net, (a) does the ball clear it and (b) what is the distance between the center of the ball and the top of the net? Suppose that, instead, the ball is served as before but now it leaves the racquet at below the horizontal. When the ball reaches the net, (c) does the ball clear it and (d) what now is the distance between the center of the ball and the top of the net?
The fast French train known as the TGV (Train à Grande Vitesse) has a scheduled average speed of . (a) If the train goes around a curve at that speed and the magnitude of the acceleration experienced by the passengers is to be limited to , what is the smallest radius of curvature for the track that can be tolerated? (b) At what speed must the train go around a curve with a radius to be at the acceleration limit?
The minute hand of a wall clock measures from its tip to the axis about which it rotates. The magnitude and angle of the displacement vector of the tip are to be determined for three time intervals. What are the (a) magnitude and (b) angle from a quarter after the hour to half past, the (c) magnitude and (d) angle for the next half hour, and the (e) magnitude and (f) angle for the hour after that?
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