Two-dimensional Motion and Relative Motion
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Relative Motion
A boat has a constant speed, its speed relative to the water has a magnitude equal to 5 m/s. The river
current has a constant speed relative to the bank equal to 3 m/s. Determine the magnitude of the velocity
of the boat relative to the river banks in the following situations:
a) The boat navigates in the direction of the current (downstream);
b) The boat navigates in the opposite direction to the current (upstream);
c) The boat navigates in the direction perpendicular to the current.
a) The boat navigates in the direction of the current (downstream);
b) The boat navigates in the opposite direction to the current (upstream);
c) The boat navigates in the direction perpendicular to the current.
A boat moving at a constant speed of 10.8 km/h wants to cross a river perpendicularly, whose waters have a
constant speed of 1.5 m/s.
a) In what direction should the pilot keep the longitudinal axis of the boat relative to normal to the current?
b) What is the speed of the boat relative to the river bank?
a) In what direction should the pilot keep the longitudinal axis of the boat relative to normal to the current?
b) What is the speed of the boat relative to the river bank?
On a windless rainy day, the rain falls vertically to the ground at a speed of 10 m/s. A car moves
horizontally with a constant speed of 72 km/h relative to the ground.
a) What is the direction of the rain relative to the car?
b) What is the speed of the rain relative to the car?
a) What is the direction of the rain relative to the car?
b) What is the speed of the rain relative to the car?
The wheel of radius R = 15 cm in the figure rolls, without slipping, parallel to a vertical
plane. The center C of the wheel has speed v = 5 m/s. What is the magnitude of the
velocity at point B in the following cases:
a) Diameter AB is normal to the running surface;
b) Diameter AB is parallel to the rolling plane.
a) Diameter AB is normal to the running surface;
b) Diameter AB is parallel to the rolling plane.
A worker holds one end of a straight board of length a, while the other end rests on a cylindrical
drum so that the board is in a horizontal position. When moving the board forward, the worker makes the
drum roll without slipping along the horizontal plane, and during the movement, the board remains
horizontal. Determine the distance d that the worker will travel until the end he is holding touches the
drum.
Two-Dimensional Motion
From the vertex of a right angle leaves, with a time interval equal to n seconds, two drivers, with
constant speeds on both sides. Calculate the speeds of the two drivers, knowing that after t second,
from the start of the second driver, its distance is d, and after T second is D.
Three identical spheres are launched from the same height H with the same speeds. Sphere A
is thrown vertically downward, B is thrown vertically upward, and C is thrown horizontally.
Which one gets to the ground at a higher speed, neglecting the resistance of the air.
A ball rolls on a horizontal table of height H, with constant speed v0,
frictionless, until it falls over the edge, calculate:
a) The time for the ball to hit the ground;
b) The horizontal distance, from the edge of the table, where the ball hits the floor;
c) The equation motion;
d) The speed with which the ball hits the ground.
a) The time for the ball to hit the ground;
b) The horizontal distance, from the edge of the table, where the ball hits the floor;
c) The equation motion;
d) The speed with which the ball hits the ground.
Solution with reference frame on the ground pointing upwards
Solution with reference frame on the table pointing downwards
Solution with reference frame on the table pointing upwars
Solution with reference frame on the table pointing downwards
Solution with reference frame on the table pointing upwars
A ball rolls on the roof of a house until it falls over the edge with speed v0. The
height of the point from which the ball falls is equal to H, and the angle of inclination of the
roof with the vertical is equal to θ, calculate:
a) The time required for the ball to hit the ground;
b) The horizontal distance from the house, where the ball hits the ground;
c) The equation of motion;
d) The speed with which the ball hits the ground.
a) The time required for the ball to hit the ground;
b) The horizontal distance from the house, where the ball hits the ground;
c) The equation of motion;
d) The speed with which the ball hits the ground.
Solution with reference frame on the ground pointing upwards
Solution with reference frame on the roof pointing downwards
Solution with reference frame on the roof pointing upwards
Solution with reference frame on the roof pointing downwards
Solution with reference frame on the roof pointing upwards
A basketball player throws the ball toward the hoop at a distance of 4.6 m making a 60° angle with the
horizontal. The basket is at a height of 3.05 m and the ball is 2.25 m from the ground when it leaves the
player's hands. Calculate the initial speed of the ball and the time spent by the ball to go from the
player's hands to the hoop. Assume acceleration due to gravity g = 10 m/s2.
Solution with frame of reference on the ground pointing upwards
Solution with frame of reference on the hands of player poihting upwards
Solution with frame of reference on the hands of player poihting upwards
A projectile is fired with an initial velocity equal to v0 and at an angle
θ0 with the horizontal, the firing point and the target are on the same horizontal
plane, and neglecting air resistance, determine:
a) The maximum height that the projectile reaches;
b) The time required to reach the maximum height;
c) The duration of the total movement;
d) The maximum horizontal range of the projectile;
e) The equation of the trajectory;
f) The angle of fire that provides the maximum range;
g) Show that shots with complementary angles have the same range;
h) The speed at any point on the trajectory;
i) The components of acceleration at any point on the trajectory.
a) The maximum height that the projectile reaches;
b) The time required to reach the maximum height;
c) The duration of the total movement;
d) The maximum horizontal range of the projectile;
e) The equation of the trajectory;
f) The angle of fire that provides the maximum range;
g) Show that shots with complementary angles have the same range;
h) The speed at any point on the trajectory;
i) The components of acceleration at any point on the trajectory.
From two points A and B, situated at a distance of 1000 m from each other, on the same
horizontal plane, two rockets are launched simultaneously. One departs from point B with an
initial velocity of 200 m/s directed upwards, and the other from point A in the direction of the
vertical passing through B, making an angle of 60° with the horizon. Determine:
a) The initial speed of rocket A so that it intercepts the second;
b) After how long does the meeting of the two rockets take place;
c) At what height does the meeting take place;
d) Check if this encounter takes place during the ascent or fall of the first rocket.
a) The initial speed of rocket A so that it intercepts the second;
b) After how long does the meeting of the two rockets take place;
c) At what height does the meeting take place;
d) Check if this encounter takes place during the ascent or fall of the first rocket.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .