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Chapter 5: Problem 1

Two bullets are fired horizontally with different velocities from the same height. Which will reach the ground first? (1) Slower one (2) Faster one (3) Both will reach simultaneously (4) Cannot be predicted

Short Answer

Expert verified
(3) Both will reach simultaneously.

Step by step solution

01

Understand the Problem

We have two bullets fired horizontally from the same height but with different velocities. We need to determine which bullet hits the ground first based on their horizontal velocities.
02

Recall Gravitational Effects

Remember that the time taken for an object to fall to the ground, when only gravity affects it, depends only on the initial height and not on its horizontal velocity.
03

Apply the Uniform Acceleration Principle

When an object is under the influence of gravity alone, it falls with a constant acceleration \( g = 9.8 \, m/s^2 \). This acceleration affects the vertical motion only, not the horizontal motion.
04

Ignore Horizontal Motion Influence

Since the bullets are fired horizontally, their horizontal velocities do not influence the time taken to hit the ground. Thus, both bullets will take the same time to reach the ground as they start from the same height.
05

Conclusion

Both bullets reach the ground at the same time, because the gravitational acceleration is the same for both, and their initial vertical velocities are zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Velocity
When an object is moving horizontally, its horizontal velocity remains constant unless acted upon by external forces. This velocity is crucial for understanding how far an object travels parallel to the ground. However, in the context of projectile motion, horizontal velocity does not affect how quickly an object falls to the ground.
  • Horizontal velocity is independent of gravitational forces which act vertically.
  • It determines the range of the projectile, not the time of flight.
For example, if you fire two bullets with different horizontal velocities from the same height, the horizontal speed doesn't alter the time they take to reach the ground. This is because gravity has no impact on horizontal velocity.
Gravitational Acceleration
Gravitational acceleration refers to the acceleration of an object due to Earth's gravity. This acceleration is approximately \( g = 9.8 \, m/s^2 \) and it affects all objects equally, regardless of their mass or horizontal velocity.
  • It acts downward, influencing only the vertical motion of an object.
  • Gravitational acceleration ensures that all objects, when dropped or projected horizontally, will accelerate towards Earth at the same rate.
In our exercise, both bullets experience the same gravitational acceleration, hence they fall to the ground simultaneously, provided other conditions such as height are identical.
Uniform Acceleration
Uniform acceleration occurs when an object's velocity changes at a constant rate. In the case of gravitational acceleration, this rate is \( 9.8 \, m/s^2 \).
  • Uniform acceleration affects the bullet's vertical motion, making it increase speed as it falls.
  • This constant acceleration does not apply horizontally, where velocity remains uniform.
Thus, whether a bullet is fired faster or slower horizontally, it falls at the same rate due to the uniform gravitational acceleration.
Free Fall
Free fall describes the motion of an object under the influence of gravity alone. In a vacuum, where air resistance is absent, all objects experience free fall in the same way.
  • During free fall, initial horizontal velocity does not affect the time to reach the ground.
  • A crucial condition of free fall is that the only force acting is gravity.
In the context of the exercise, both bullets are in free fall as they are only subject to gravitational forces, leading to them reaching the ground simultaneously, despite differing horizontal velocities.

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Most popular questions from this chapter

A particle is projected with a velocity \(v\) so that its range on a horizontal plane is twice the greatest height attained. If \(g\) is acceleration due to gravity, then its range is (1) \(\frac{4 v^{2}}{5 g}\) (2) \(\frac{4 g}{5 v^{2}}\) (3) \(\frac{4 v^{3}}{5 g^{2}}\) (4) \(\frac{4 v}{5 g^{2}}\)

A shell fired from the ground is just able to cross horizontily the top of a wall \(90 \mathrm{~m}\) away and \(45 \mathrm{~m}\) high. The direction of projection of the shell will be (1) \(25^{\circ}\) (2) \(30^{\circ}\) (3) \(60^{\circ}\) (4) \(45^{\circ}\)

A helicopter is flying at \(200 \mathrm{~m}\) and flying at \(25 \mathrm{~m} \mathrm{~s}^{-1}\) at an angle \(37^{\circ}\) above the horizontal when a package is dropped from it. If the helicopter flies at constant velocity, find the \(x\) and coordinates of the location of the helicopter when the \(v\) package lands. (1) \(160 \mathrm{~m}, 320 \mathrm{~m}\) (2) \(100 \mathrm{~m}, 200 \mathrm{~m}\) (3) \(200 \mathrm{~m}, 400 \mathrm{~m}\) (4) \(50 \mathrm{~m}, 100 \mathrm{~m}\)

A ball is projected horizontally from an incline so as strike a cart sliding on the incline. Neglect height of can and point of projection of ball above incline. At the instance the ball is thrown, the speed of cart is \(v\) (in \(\mathrm{m} / \mathrm{s}\) ). Find \(v\) that the ball strikes the cart.

Projectile motion is a combination of twe enslonal motions: one in horizontal and other in vertical direction. Motion in \(2 D\) means in a plane. Necessary condition for \(2 D\) motion is that the velocity vector is coplanar to the acceleration vector. In case of projectile motion, the angle between velocity and acceleration will be \(0^{\circ}<\theta<180^{\circ} .\) During the projectile motion, the horizontal component of velocity remains unchanged but vertical component of velocity is time dependent. Now answer the following questions: An object is projected from origin in \(x-y\) plane in whict velocity changes according to relation \(\vec{v}=a \breve{i}+b x \hat{j}\). Pat of particle is (1) Hyperbolic (2) Circular (3) Elliptical (4) Parabolic
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