Projectile Motion with Collision of Projectile

Projectile motion is a form of motion experienced by an object that is thrown near the Earth's surface and moves along a curved path under the action of gravity only. When a projectile collides with another object during its trajectory, the situation becomes more complex, involving both the principles of projectile motion and collision dynamics.

Understanding Projectile Motion

Before delving into collisions, let's first understand the basics of projectile motion. The motion of a projectile is characterized by two components:

1. Horizontal Motion: The horizontal motion is uniform, meaning the projectile moves at a constant horizontal velocity because there are no horizontal forces acting on it (ignoring air resistance).

2. Vertical Motion: The vertical motion is uniformly accelerated, meaning the only force acting on the projectile is gravity, which causes the projectile to accelerate downwards at a constant rate.

The key equations for projectile motion are:

• Horizontal displacement: $x = v_{0x} t$
• Vertical displacement: $y = v_{0y} t - \frac{1}{2} g t^2$
• Horizontal velocity: $v_{x} = v_{0x}$
• Vertical velocity: $v_{y} = v_{0y} - g t$

where $v_{0x}$ and $v_{0y}$ are the initial horizontal and vertical velocities, $g$ is the acceleration due to gravity, and $t$ is the time.

Collision of Projectile

When a projectile collides with another object, the laws of conservation of momentum and energy (in elastic collisions) come into play. The collision can be elastic or inelastic:

• Elastic Collision: Both momentum and kinetic energy are conserved.
• Inelastic Collision: Only momentum is conserved; kinetic energy is not conserved.

Important Points to Consider

• Momentum Conservation: The total momentum before the collision is equal to the total momentum after the collision.
• Energy Conservation: In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision.
• Collision Angle: The angle at which the collision occurs can significantly affect the outcome of the collision.

Formulas for Collision

For a two-body collision, the conservation of momentum can be expressed as:

• $m_1 \vec{v}{1i} + m_2 \vec{v}{2i} = m_1 \vec{v}{1f} + m_2 \vec{v}{2f}$

where $m_1$ and $m_2$ are the masses of the two objects, $\vec{v}{1i}$ and $\vec{v}{2i}$ are the initial velocities, and $\vec{v}{1f}$ and $\vec{v}{2f}$ are the final velocities.

For an elastic collision, the conservation of kinetic energy is given by:

• $\frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

Differences and Important Points

Aspect	Projectile Motion Only	Projectile Motion with Collision
Forces Involved	Gravity only	Gravity and contact forces
Motion Type	Curved path	Curved path followed by interaction
Conservation Laws	Not applicable	Momentum and possibly energy
Complexity	Moderate	High
Equations Used	Kinematic equations	Kinematic + conservation equations
Outcome	Predictable trajectory	Trajectory altered by collision

Examples

Example 1: Projectile Motion without Collision

A ball is thrown with an initial velocity of 20 m/s at an angle of 30 degrees above the horizontal. Calculate the maximum height reached by the ball.

Solution:

Using the vertical motion equation:

• $v_{0y} = v_0 \sin(\theta) = 20 \sin(30^\circ) = 10$ m/s
• $y_{max} = \frac{v_{0y}^2}{2g} = \frac{10^2}{2 \times 9.8} \approx 5.1$ m

Example 2: Projectile Motion with Elastic Collision

A projectile of mass $m_1$ collides elastically with a stationary object of mass $m_2$ at the highest point of its trajectory. If $m_1 = m_2$ and the projectile was moving horizontally at the time of collision, find the velocities of both objects after the collision.

Solution:

Using conservation of momentum and kinetic energy:

• $m_1 v_{1i} = m_1 v_{1f} + m_2 v_{2f}$
• $\frac{1}{2} m_1 v_{1i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

Since $m_1 = m_2$ and the second object is initially at rest:

• $v_{1i} = v_{1f} + v_{2f}$
• $v_{1i}^2 = v_{1f}^2 + v_{2f}^2$

Solving these equations, we find that $v_{1f} = 0$ and $v_{2f} = v_{1i}$. The first object comes to rest, and the second object moves with the initial velocity of the first object.

Projectile motion with collision adds a layer of complexity to the analysis of an object's motion. Understanding the principles of both projectile motion and collision dynamics is crucial for solving problems involving such scenarios.