Kinematics (1D) and describing trajectories

Describing trajectories

Problem:

A circle of radius a rolls on a straight line in the positive x-direction.  The trajectory y(x) of a given point P on this circle is a cycloid.

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(a)  Find the parametric representation x(y) of this cycloid.
(b)  Find the length of the path of the point P, when the circle has completed one revolution, i.e. when the center of the circle has traveled a distance 2πa.

Solution:


Kinematics (1D)

Problem:

A chipmunk is taking a sunbath 5.0 m away from its burrow.  It then decides to go jogging.  The chipmunk takes off and runs directly away from the burrow, so that its velocity is inversely proportional to the distance from the burrow.  If the initial speed of the chipmunk is 2.0 m/s, how long would it take it to run 15.0 m?

Solution:

Problem:

Laurel running at 10 m/s is 40 m behind Hardy when Hardy starts from rest on his moped with an acceleration 1.0 m/s2.  How long does it take for Laurel to catch up with Hardy?

Solution:

Problem:

You are traveling east at 30 miles per hour.  You see a ball rolling onto the road and you break hard, because you are afraid that a child will come running after the ball.  You come to a stop in 0.8 seconds.  What is the direction of your average velocity in this short time interval?  What is your average acceleration?

Solution:

Problem:

A mass m starting from rest slides without friction in the field of gravity from the top of a vertical diameter d of a circle, A, along a straight wire connecting A and a point on the circumference, B.  The wire makes an angle θ with the diameter, as shown in the figure.  Does the time it takes to slide from A to B depend on θ?  Justify your answer.

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Solution:

Problem:

Two cars approach an intersection of two perpendicular roads as shown.  The velocities of the cars are v1 and v2
At the moment when car 1 reaches the intersection, the separation between the cars is d.
What is the minimum separation between the cars during this motion?

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Solution:

Problem:

A prankster drops a water-filled balloon out of a window.  The balloon is released from rest at a height of 10 m above the ears of an innocent man who is the target.  Then, because of a guilty conscience, the prankster shouts a warning after the balloon is released.  The warning will do no good, however, if shouted after the balloon reaches a certain point, even if the man could react infinitely quickly.  Ignoring the effect of air resistance on the balloon, determine how far above the innocent man's ears this point is.  The speed of sound is 340 m/s.

Solution:

Problem:

A boy wants to throw a can straight up and then hits it with a second can.  He wants the collision to occur 4 m above the throwing point.  In addition, he knows that the time he needs between throws is 3.0 second.  Assuming that he throws the second can 3.0 seconds after he throws the first can and that he throws both cans with the same speed, what must the initial speed be?  What are the speeds of both cans when they collide?

Solution:

Problem:

A ball that is thrown upward near the surface of the earth with a velocity of 50 m/s will come to rest about 5 second later.  If the ball were thrown up with the same initial velocity on Planet X, after 5 seconds it would still be moving upwards at nearly 31 m/s.  The magnitude of the gravitational field near the surface of Planet X is what fraction of the gravitational field near the surface of the earth?

Solution:

Problem:

A projectile launched from the ground explodes into three fragments of equal mass at the top point of the trajectory.  One of the fragments lands t seconds after the explosion; two other fragments land simultaneously 2t seconds after the explosion.  How high above the ground does the projectile explode?

Solution:

Problem:

A student in a boat wants to cross the river from point K to point L (see the diagram).  The current of the river is v = 2.0 km/h; the speed of the boat in still water is u = 5.0 km/h.  The river is a = 0.25 km wide; distance LM is b = 0.5 km (see the diagram).  How long would the trip take?

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Solution:

Problem:

Two ferries leave from opposite sides of a river.  They move with different speeds and meet 400 m from the left bank.  Each continues to its destination, take 5 minutes to unload and load, and then starts the return trip.  The ferries meet again 200 m from the right bank.  How wide is the river?

Solution:

Kinematics (electric force )