Question

A projectile is launched from a height \(y_{0}=0\). For a given launch angle, if the launch speed is doubled, what will happen to the range \(R\) and the maximum height \(H\) of the projectile? a) \(R\) and \(H\) will both double. b) \(R\) and \(H\) will both quadruple. c) \(R\) will double, and \(H\) will stay the same. d) \(R\) will quadruple, and \(H\) will double. e) \(R\) will double, and \(H\) will quadruple.

Short answer

a) Both \(R\) and \(H\) will double. b) Both \(R\) and \(H\) will quadruple. c) \(R\) will double, and \(H\) will quadruple. d) \(R\) will quadruple, and \(H\) will double. Answer: d) \(R\) will quadruple, and \(H\) will double.

Step-by-step solution

Recall the equations for projectile motion

In this problem, we'll be using the equations \(R = \frac{v_{0}^{2}\sin(2\theta)}{g}\), where \(R\) is the range and the maximum height of the projectile, \(H = \frac{v_{0}^{2}\sin^2(\theta)}{2g}\), where \(v_{0}\) is the initial launch speed, \(\theta\) is the launch angle, and \(g\) is the gravitational acceleration.

Double the launch speed in the equations

To determine what happens to the range and maximum height when the launch speed doubles, we will substitute \(2v_{0}\) for \(v_{0}\) in both equations. For the range: \(R' = \frac{(2v_{0})^{2}\sin(2\theta)}{g} = 4\frac{v_{0}^{2}\sin(2\theta)}{g} = 4R\). For the maximum height: \(H' = \frac{(2v_{0})^{2}\sin^2(\theta)}{2g} = 4\frac{v_{0}^{2}\sin^2(\theta)}{2g} = 2H\).

Choose the answer that matches the results

From the previous step, we find that when the launch speed is doubled, the range R will quadruple and the maximum height H will double. So, the correct answer is: d) \(R\) will quadruple, and \(H\) will double.

Key concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without reference to the causes of this motion. It involves the description of motion in terms of displacement, velocity, and acceleration. For projectile motion, we focus on objects that are launched into the air and are subject to gravitational acceleration, but no other forces. To analyze such motion, we use kinematic equations that describe the vertical and horizontal aspects separately. Since gravity acts downwards, the vertical motion is affected, while the horizontal motion remains constant (disregarding air resistance).

In our exercise, the projectile's launch speed and angle determine both its maximum height and range. The kinematic equations used in projectile motion allow us to predict how changing the initial conditions, such as doubling the launch speed, will alter the trajectory and final distance traveled.

Range of a Projectile

The range of a projectile is the horizontal distance it travels from the launch point to the point where it strikes the ground. The formula for the range is derived from the kinematic principles and is influenced by the projectile's initial velocity, the angle of launch, and the gravitational acceleration. The equation is:
\[\begin{equation}R = \frac{v_{0}^{2}\times \text{sin}(2\theta)}{g}\right\text{-end},\text{where}\(v_{0})\)is\)the\)initial\)launch\)speed,\)(\theta\)is\)the\)launch\)angle,\)and\)(g\)is\)the\)gravitational\)acceleration.\) Doubling the launch speed has a profound effect on the range. As our exercise demonstrates, if you double\)\text{sin}\)\text{v_{0}}, the range\)(R\)increases by a factor of four, not two. This is due to the range being proportional to the square of the initial velocity.

Maximum Height of a Projectile

The maximum height of a projectile is the highest vertical position it reaches in its flight. It is determined by three factors: the initial speed of the projectile, the angle of projection, and the acceleration due to gravity. The formula for the maximum height is given by:
\[\begin{equation}H = \frac{v_{0}^{2}\times \text{sin}^2(\theta)}{2g}\right\text{-end},\text{where}\)\text{onclusively}}\)are\)the\)same\)variables\)defined\)previously.If\)the\)launch\)speed\)is\)doubled,\)as\)illustrated\)in\)our\)example,\)the\)maximum\)height\)(H\)will\)increase\)by\)a\)factor\)of\)two. This is because the maximum height is directly proportional to the square of the initial speed (and since\)(2^2\)=\)4\),\)we\)end\)up\)with\)twice\)the\)original\)height, not four times.

Gravitational Acceleration

Gravitational acceleration (g) is a constant that measures the acceleration due to gravity at the Earth's surface. It has a standard value of approximately 9.81 m/s^2. This acceleration pulls objects toward the Earth and plays a significant role in projectile motion. It only affects the vertical component of the motion, causing the projectile to slow down as it reaches its peak height and speed up as it falls back down. Gravitational acceleration is the reason why projectiles have a parabolic trajectory. Regardless of the horizontal velocity, all projectiles (neglecting air resistance) accelerate towards the Earth at the same rate, which has crucial implications in calculations for range and maximum height as seen in our textbook exercise. Doubling the launch speed doesn't change the value of (g) but increases the effect that this acceleration has on the projectile's motion.