Problem 47 - Master Resource Book in JEE Main Physics

Question

If a body is projected with an angle to the horizontal, then \(\quad\) [EAMCET 2008] (a) its velocity is always perpendicular to its acceleration (b) its velocity becomes zero at maximum height (c) its velocity makes zero angle with the horizontal at its maximum height (d) the body just before hitting the ground, the direction of velocity coincides with the acceleration

Step-by-Step Solution

Answer

(c) at maximum height, the velocity is horizontal (zero angle with horizontal).

Step 1: Understand the Problem

The problem deals with a projectile motion, which means a body is thrown with an initial velocity at a certain angle to the horizontal. We need to find out which statement about the velocity and acceleration of the body is true when it is projected.

Step 2: Analyze Statement (a)

Statement (a) claims the body's velocity is always perpendicular to its acceleration. In projectile motion, the acceleration due to gravity acts downwards, but the velocity has both horizontal and vertical components. These are not always perpendicular; thus statement (a) is false.

Step 3: Analyze Statement (b)

Statement (b) states the velocity becomes zero at maximum height. At maximum height, the vertical component of velocity is zero, but the horizontal component is still present, so the total velocity is not zero. This makes statement (b) false.

Step 4: Analyze Statement (c)

Statement (c) suggests the velocity makes a zero angle with the horizontal at its maximum height. This means the velocity is horizontal at maximum height, which is true because the vertical component is zero at this point, leaving only the horizontal component.

Step 5: Analyze Statement (d)

Statement (d) claims that just before hitting the ground, the direction of velocity coincides with the acceleration. However, the acceleration remains vertical (downward due to gravity) throughout the motion. The velocity at any point, including just before hitting the ground, has a horizontal component, so the directions cannot coincide. Therefore, statement (d) is false.

Step 6: Conclusion

Based on analyzing each statement, statement (c) is correct as it properly describes the velocity condition at the maximum height in projectile motion.

Key Concepts

Velocity Components in Projectile Motion

In projectile motion, understanding the components of velocity is crucial. When a body is projected with an initial velocity at an angle to the horizontal, this velocity can be broken down into two components: horizontal and vertical.

• **Horizontal Component**: This is constant since no external forces like air resistance are typically considered in basic projectile motion problems. Its formula is given by \( v_x = v_0 \cdot \cos(\theta) \), where \( v_0 \) is the initial velocity and \( \theta \) is the angle of projection.
• **Vertical Component**: This changes with time due to gravity, calculated using \( v_y = v_0 \cdot \sin(\theta) - g \cdot t \). Here, \( g \) represents the acceleration due to gravity, usually \( 9.81 \text{ m/s}^2 \). The initial vertical speed decreases as it travels upwards until it reaches zero at the maximum height.

At any point in time, these components combine to give the resultant velocity of the projectile, but they are treated separately to simplify calculations.

Acceleration due to Gravity

Acceleration due to gravity is a fundamental concept in projectile motion. It is the force that pulls objects back to Earth, affecting the vertical component of their motion. Here are some key points:

• The acceleration due to gravity is denoted by \( g \), typically \( 9.81 \text{ m/s}^2 \), and acts downward towards the center of the Earth.
• In projectile motion, this acceleration is the reason why the vertical component of the velocity changes with time. As an object ascends, gravity reduces the vertical velocity until it momentarily becomes zero at the maximum height. As it descends, gravity increases the vertical velocity.
• The horizontal component of the projectile's velocity remains unaffected by gravity, assuming no other forces act on the object.

Overall, gravity plays a pivotal role in shaping the trajectory of the projectile, instigating a parabolic path.

Maximum Height in Projectile Motion

The maximum height in projectile motion is where the projectile reaches its highest vertical position. At this point, the vertical component of the velocity becomes zero, offering a unique brief moment of pause before descent.To calculate the maximum height, we use the formula:\[ h_{max} = \frac{(v_0 \cdot \sin(\theta))^2}{2g} \]Here, \( v_0 \) is the initial velocity, \( \theta \) is the launch angle, and \( g \) is the acceleration due to gravity. Key insights include:

• At maximum height, the entire velocity is horizontal. This is why option (c) in the exercise is correct, as the angle between the velocity and the horizontal is zero.
• This peak point also divides the projectile's motion into two symmetric halves: ascent and descent.

Understanding how to derive and use this formula helps in predicting how high a projectile will travel and when during its flight the maximum height will occur.