Resource Lesson
Chase Problems: Projectiles
Printer Friendly Version
s
_{pursuer}
= "gap" + s
_{leader}
v
_{o}
t + ½at
^{2}
number
v
_{o}
t + ½at
^{2}
vt
vt
Each column in the above table states the allowed behaviors for the pursuer and the leader. Each participant can either be experiencing accelerated or linear motion. The numerical value of the "gap" can be equal to zero (if the two objects start side-by-side) or it can be a nonzero number. The parameter
t
, for time, unites the equations. To solve chase equations, you first determine the time that is required for the two objects to come together - then, you use that time to determine the position of their collision.
To work this type of problem, one object is considered the leader and the other is the pursuer. The pursuer, in reaching the leader's final location, must not only close the leader's original gap but also account for any subsequent displacement the leader travels while being chased.
Example
A ball of mass 1.0 kg is dropped from rest from a height of 5.0 meters above the ground, as shown in the diagram at the right. It undergoes a perfectly elastic collision with the ground [that is, it leaves the surface of the ground with exactly the same amount of kinetic energy (KE = ½mv
^{2}
) with which it initially strikes the ground] and rebounds at a speed of 9.9 m/sec. At the instant that the ball rebounds, a small blob of clay of mass 0.1 kg is released from rest from the original height H, directly above the ball, as shown. The clay blob while descending eventually collides with the ascending ball. Assume that air resistance is negligible.
To determine how much time passes after the clay blob is released until it strikes the ball we must first set up the chase equation. Let's assume that the 1.0-kg ball is the pursuer (since it is traveling in a positive direction) and that the 0.1-kg clay blob is the leader.
s
_{pursuer}
= "gap" + s
_{leader}
v
_{o}
t + ½at
^{2}
number
v
_{o}
t + ½at
^{2}
vt
vt
s > 0
s < 0
9.9t + ½(-9.8)t
^{2}
5 meters
0t + ½(-9.8)t
^{2}
9.9t + ½(-9.8)t
^{2}
= 5 + [0t + ½(-9.8)t
^{2}
]
9.9t - 4.9t
^{2}
= 5 - 4.9t
^{2}
9.9t = 5
t = 0.505 seconds
Next let's determine the height above the ground at which the collision takes place. Since this represents the distance that the ball traveled upwards after its bounce, we can substitute the time, 0.505 seconds, into the ball's equation.
s = 9.9t + ½(-9.8)t
^{2}
s = 9.9(0.505) + ½(-9.8)(0.505)
^{2}
s = 3.7 meters
The position-time graph for this problem would look like the one shown below. Note that the two parabolas intersect at a height of 3.7 meters and a time of 0.505 seconds. The purple parabola represents the clay blob which is gaining speed in a negative direction while the blue parabola represents the ball which is losing speed while traveling in a positive direction.
Next, let's determine how fast the ball and the clay were traveling at the moment that they collided.
clay (descending)
ball (ascending)
v
_{f}
= v
_{o}
+ at
v
_{f}
= v
_{o}
+ at
v
_{f}
= 0 +(-9.8)(0.505)
v
_{f}
= 9.9 + (-9.8)(0.505)
v
_{f}
= -4.95 m/sec
v
_{f}
= +4.95 m/sec
The velocity-time graph for this problem is shown below. Each line has the same slope of -9.8 m/sec
^{2}
. The blue line starts at 9.9 m/sec and ends at 4.95 m/sec while the purple line starts are 0 m/sec and ends at -4.95 m/sec. Note that the blue area between the line and the x-axis must equal 3.7 meters while the purple area between the line and the x-axis must equal 5.0 - 3.7 or 1.3 meters. The combined blue and purple areas must equal the entire original gap, 5.0 meters.
Practice your skills with this type of problem by completing the accompanying worksheet entitled
Chase Problems: Projectiles
.
Related Documents
Lab:
Labs -
A Photoelectric Effect Analogy
Labs -
Acceleration Down an Inclined Plane
Labs -
Ballistic Pendulum: Muzzle Velocity
Labs -
Coefficient of Friction
Labs -
Coefficient of Kinetic Friction (pulley, incline, block)
Labs -
Collision Pendulum: Muzzle Velocity
Labs -
Conservation of Momentum
Labs -
Cookie Sale Problem
Labs -
Flow Rates
Labs -
Freefall Mini-Lab: Reaction Times
Labs -
Freefall: Timing a Bouncing Ball
Labs -
Galileo Ramps
Labs -
Gravitational Field Strength
Labs -
Home to School
Labs -
InterState Map
Labs -
LAB: Ramps - Accelerated Motion
Labs -
LabPro: Newton's 2nd Law
Labs -
LabPro: Uniformly Accelerated Motion
Labs -
Mass of a Rolling Cart
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Monkey and the Hunter Animation
Labs -
Monkey and the Hunter Screen Captures
Labs -
Projectiles Released at an Angle
Labs -
Ramps: Sliding vs Rolling
Labs -
Range of a Projectile
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Rube Goldberg Challenge
Labs -
Target Lab: Ball Bearing Rolling Down an Inclined Plane
Labs -
Terminal Velocity
Labs -
Video LAB: A Gravitron
Labs -
Video Lab: Ball Bouncing Across a Stage
Labs -
Video LAB: Ball Re-Bounding From a Wall
Labs -
Video Lab: Cart Push #2 and #3
Labs -
Video Lab: Falling Coffee Filters
Labs -
Video Lab: Two-Dimensional Projectile Motion
Resource Lesson:
RL -
Accelerated Motion: A Data Analysis Approach
RL -
Accelerated Motion: Velocity-Time Graphs
RL -
Analyzing SVA Graph Combinations
RL -
Average Velocity - A Calculus Approach
RL -
Chase Problems
RL -
Comparing Constant Velocity Graphs of Position-Time & Velocity-Time
RL -
Constant Velocity: Position-Time Graphs
RL -
Constant Velocity: Velocity-Time Graphs
RL -
Derivation of the Kinematics Equations for Uniformly Accelerated Motion
RL -
Derivatives: Instantaneous vs Average Velocities
RL -
Directions: Flash Cards
RL -
Freefall: Horizontally Released Projectiles (2D-Motion)
RL -
Freefall: Projectiles in 1-Dimension
RL -
Freefall: Projectiles Released at an Angle (2D-Motion)
RL -
Monkey and the Hunter
RL -
Summary: Graph Shapes for Constant Velocity
RL -
Summary: Graph Shapes for Uniformly Accelerated Motion
RL -
SVA: Slopes and Area Relationships
RL -
Vector Resultants: Average Velocity
Review:
REV -
Test #1: APC Review Sheet
Worksheet:
APP -
Hackensack
APP -
The Baseball Game
APP -
The Big Mac
APP -
The Cemetary
APP -
The Golf Game
APP -
The Spring Phling
CP -
2D Projectiles
CP -
Dropped From Rest
CP -
Freefall
CP -
Non-Accelerated and Accelerated Motion
CP -
Tossed Ball
CP -
Up and Down
NT -
Average Speed
NT -
Back-and-Forth
NT -
Crosswinds
NT -
Headwinds
NT -
Monkey Shooter
NT -
Pendulum
NT -
Projectile
WS -
Accelerated Motion: Analyzing Velocity-Time Graphs
WS -
Accelerated Motion: Graph Shape Patterns
WS -
Accelerated Motion: Practice with Data Analysis
WS -
Advanced Properties of Freely Falling Bodies #1
WS -
Advanced Properties of Freely Falling Bodies #2
WS -
Advanced Properties of Freely Falling Bodies #3
WS -
Average Speed and Average Velocity
WS -
Average Speed Drill
WS -
Charged Projectiles in Uniform Electric Fields
WS -
Chase Problems #1
WS -
Chase Problems #2
WS -
Chase Problems: Projectiles
WS -
Combining Kinematics and Dynamics
WS -
Constant Velocity: Converting Position and Velocity Graphs
WS -
Constant Velocity: Position-Time Graphs #1
WS -
Constant Velocity: Position-Time Graphs #2
WS -
Constant Velocity: Position-Time Graphs #3
WS -
Constant Velocity: Velocity-Time Graphs #1
WS -
Constant Velocity: Velocity-Time Graphs #2
WS -
Constant Velocity: Velocity-Time Graphs #3
WS -
Converting s-t and v-t Graphs
WS -
Energy Methods: More Practice with Projectiles
WS -
Energy Methods: Projectiles
WS -
Force vs Displacement Graphs
WS -
Freefall #1
WS -
Freefall #2
WS -
Freefall #3
WS -
Freefall #3 (Honors)
WS -
Horizontally Released Projectiles #1
WS -
Horizontally Released Projectiles #2
WS -
Kinematics Along With Work/Energy
WS -
Kinematics Equations #1
WS -
Kinematics Equations #2
WS -
Kinematics Equations #3: A Stop Light Story
WS -
Lab Discussion: Gravitational Field Strength and the Acceleration Due to Gravity
WS -
Position-Time Graph "Story" Combinations
WS -
Projectiles Released at an Angle
WS -
Rotational Kinetic Energy
WS -
SVA Relationships #1
WS -
SVA Relationships #2
WS -
SVA Relationships #3
WS -
SVA Relationships #4
WS -
SVA Relationships #5
WS -
Work and Energy Practice: An Assortment of Situations
TB -
2A: Introduction to Motion
TB -
2B: Average Speed and Average Velocity
TB -
Antiderivatives and Kinematics Functions
TB -
Honors: Average Speed/Velocity
TB -
Kinematics Derivatives
TB -
Projectile Summary
TB -
Projectile Summary
TB -
Projectiles Mixed (Vertical and Horizontal Release)
TB -
Projectiles Released at an Angle
TB -
Set 3A: Projectiles
PhysicsLAB
Copyright © 1997-2024
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton