Kinematics and Projectile Motion: Problem Set Study Guide

Kinematics and Projectile Motion: Problem Set Study Guide

1. Pursuit Problems with Constant Acceleration

These problems involve analyzing the motion of two objects, typically one starting from rest and accelerating, while the other moves at a constant speed. The goal is to determine when and where the accelerating object catches up to the other.

• Constant Acceleration: When an object accelerates at a constant rate, its velocity and position as functions of time are given by:

  • Velocity: $v = v_0 + at$

  • Position: $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• Relative Motion: To find when one object overtakes another, set their positions equal and solve for time.

• Example Application: A police officer starts from rest and accelerates at $3.0\ \mathrm{m/s^2}$ to catch a car moving at $30\ \mathrm{m/s}$.

2. Overtaking and Relative Motion at Intersections

These problems involve two vehicles starting at different times or with different accelerations, and require determining when and where one overtakes the other.

• Key Equations: Use the kinematic equations for each vehicle, considering their initial velocities and accelerations.

• Example: An automobile starts from rest with $a = 2.50\ \mathrm{m/s^2}$, and a truck passes with $v = 15.0\ \mathrm{m/s}$.

• Steps:

  1. Write position equations for both vehicles.

  2. Set positions equal to solve for the time of overtaking.

  3. Substitute back to find the distance and velocities at that instant.

3. Free Fall and Vertical Motion

Problems involving objects thrown vertically upward or dropped from a height, under the influence of gravity.

• Acceleration due to Gravity: $g = 9.8\ \mathrm{m/s^2}$ (downward)

• Key Equations:

  • Upward motion: $v = v_0 - gt$

  • Height: $y = y_0 + v_0 t - \frac{1}{2} g t^2$

• Time to Reach Maximum Height: Set $v = 0$ and solve for $t$.

• Time to Hit the Ground: Set $y = 0$ and solve for $t$.

• Example: A ball is thrown straight up at $10\ \mathrm{m/s}$ from a $29.4\ \mathrm{m}$ building.

4. Simultaneous Vertical Motions

Problems where two objects are released simultaneously from different heights or with different initial velocities.

• Key Concept: Set the position equations for both objects and solve for the time and position where they meet.

• Example: Ball 1 is dropped from $24\ \mathrm{m}$, Ball 2 is thrown upward from the ground at $12\ \mathrm{m/s}$.

5. Projectile Motion

Projectile motion involves two-dimensional motion under gravity, with an initial velocity at an angle to the horizontal.

• Horizontal Motion: $x = v_0 \cos\theta \cdot t$ (no horizontal acceleration)

• Vertical Motion: $y = y_0 + v_0 \sin\theta \cdot t - \frac{1}{2} g t^2$

• Maximum Height: Occurs when vertical velocity is zero.

• Time of Flight: Total time the projectile is in the air.

• Example: Boulder ejected upward from a volcano at $40\ \mathrm{m/s}$, then moves under gravity.

6. Horizontal Projectile Motion

Objects projected horizontally from a height follow a curved path due to gravity.

• Time to Fall: Determined by vertical motion: $y = y_0 - \frac{1}{2} g t^2$

• Horizontal Distance: $x = v_0 t$

• Example: A physics book slides off a table at $1.10\ \mathrm{m/s}$ and hits the floor in $0.35\ \mathrm{s}$.

7. Projectile Kicks and Goalposts

Analyzing the trajectory of a ball kicked at an angle to clear a horizontal bar (goalpost).

• Key Steps:

  1. Resolve initial velocity into horizontal and vertical components.

  2. Calculate time to reach the horizontal distance.

  3. Determine vertical position at that time to see if it clears the bar.

• Example: Football kicked at $20\ \mathrm{m/s}$ at $30^\circ$ to clear a $3.05\ \mathrm{m}$ bar $20\ \mathrm{m}$ away.

8. Motion Down an Incline with Constant Acceleration

Objects rolling or sliding down an incline experience constant acceleration due to gravity's component along the slope.

• Key Equations:

  • Displacement: $x = \frac{1}{2} a t^2$ (if starting from rest)

  • Velocity: $v = a t$

• Example: Boulder rolls down a hill, traveling $2\ \mathrm{m}$ in the first second.

Summary Table: Kinematic Equations

Equation	Variables	Use
$v = v_0 + at$	v: final velocity, v_0: initial velocity, a: acceleration, t: time	Find velocity after time t
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	x: position, x_0: initial position, v_0: initial velocity, a: acceleration, t: time	Find position after time t
$v^2 = v_0^2 + 2a(x - x_0)$	v: final velocity, v_0: initial velocity, a: acceleration, x: position, x_0: initial position	Find velocity or position without time

Additional info:

• All problems assume negligible air resistance unless otherwise stated.

• For projectile motion, the horizontal and vertical motions are independent except for the time variable.