Kinematics (projectile motion)

Problem:

Two trucks are parked back to back in opposite directions on a straight, horizontal road.  The trucks quickly accelerate simultaneously to 3.0 m/s in opposite directions and maintain these velocities.  When the backs of the trucks are 20 meters apart, a boy in the back of one truck throws a stone at an angle of 40 degrees above the horizontal at the other truck.  How fast must he throw, relative to the truck, if the stone is to just land in the back of the other truck?

Solution:

Problem:

A player tossed a ball at some angle relative to the horizon.  The maximum speed of the ball during the flight was 12 m/sec and the minimum speed was 6 m/sec.  What was the maximum height of the ball during the flight?  Please neglect air resistance.

Solution:

Problem:

A rock is launched from the ground level at a speed v directed at an angle θ with the horizontal.  It is noticed that some (unknown) time t after the launch, the distance between the rock and the launch point begins to decrease.
(a)  Find the smallest launch angle θ consistent with this observation.
(b)  Find t, neglecting the air resistance.

Solution:

Problem:

Two points, A and B, are located on the ground a certain distance d apart.  Two rocks are launched simultaneously from points A and B, with equal speeds but at different angles.  Each rock lands at the launch point of the other.  Knowing that one of the rocks is launched at an angle θ0 > 45o, find the minimum distance between the rocks during the flight in terms of d and θ0?

Solution:

Problem:

A catapult set on the ground can launch a rock a maximum horizontal distance L.  What would be the maximum horizontal launch distance if the catapult is set on a platform moving forward with constant speed equal to the launch speed of the rock?
Neglect the air resistance and assume that the rock is launched from the ground level in both cases.

Solution:

Problem:

Consider a lawn sprayer consisting of a spherical cap (α0 = 45o) provided with a large number of equal holes through which water is ejected with speed v0.  The lawn is not evenly sprayed if the holes are evenly spaced.  How must the number of holes per unit area, r(α), be chosen to achieve uniform spraying of a circular area?  Assume the radius of the sprinkler cap is very much less than the radius of the area to be sprayed, and the surface of the cap is at the level of the lawn.

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Solution:

Problem:

A particle is launched with velocity v = 2 ft/s j along the ridge of a roof, but the equilibriums is unstable and it immediately starts accelerating down the right side of the roof.  Using the coordinate
system in the figure and neglecting friction, what are the particle's x- and y-coordinates when it hits the ground?
Use the following measurements:  |
Roof Rise: zR = 4 ft; Roof Width: xR= 24 ft;  Roof Length: yR = 24 ft; 
Wall Height: zW = 8 ft;  g = 32 ft/s2.
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Solution:
 

Problem:

In a carnival game, you have to throw a ball with speed v0 at an angle θ in order to hit a target on the other side of the platform, located a distance h away.  The platform is inclined at an angle φ.  Find the angle θ in terms of the other variables.  

Solution:

Problem:

A tree trunk lies on the ground.  The trunk has a shape of a cylinder with radius R.  A flea attempts to jump over the trunk.  What is the minimum initial speed v that enables the flea to reach the other side?  Assume that the flea is intelligent enough to select the optimal take-off point on the ground.
Consider two cases.
(a)  The flea is allowed to slide on the frictionless trunk.
(b)  The flea is not allowed to slide on the trunk and must clear the trunk.

Solution:

Problem:

A marble bounces down stairs in a regular manner, hitting each step at the same place and bouncing the same height above each step. 

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The stair height equals its depth (tread = rise) and the coefficient of restitution ε is given.  Find the necessary horizontal velocity and bounce height.  
(The coefficient of restitution is defined as ε = -vf/vi, where vf and vi are the vertical velocities just after and before the bounce. respectively).

Solution:

Problem:

(a)  An astronaut on a strange planet finds that she can jump a maximum horizontal distance of 15 m if her initial speed is 3 m/s.  What is the free-fall acceleration on the planet?
(b)  How much work is required to raise a 100 g block to a height of 200 cm and simultaneously give it a velocity of 300 cm/sec?

Problem:

An plane, inclined at θ = 20o, touches a wall as shown in the picture.  You drop a small, perfectly elastic ball from a height h = 1.5 m onto the onto the plane.  The ball falls from rest.  You do not move your hand.  At what distance d from the wall do you have to drop it so that it bounces back into your hand?

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Problem:

A wheel of radius b is rolling along a muddy road with a speed v.  Particles of mud attached to the wheel are being continuously thrown off from all points of the wheel.  If v2 > 2bg, where g is the acceleration of gravity, find the maximum height above the road attained by the mud, H = H(b,v,g).