91Ó°ÊÓ

Vertical motion involves the movement of an object in a straight line perpendicular to the horizontal component, primarily influenced by forces such as gravity. In the context of our exercise, a missile is launched vertically from a moving cart. The initial vertical velocity given is \(40 \ \mathrm{m/s}\), which impacts the motion wholly separate from the horizontal motion of the cart.

To calculate how high the rocket travels, we apply the vertical motion formula:

• Initial vertical velocity \( v_{0y} = 40.0 \ \mathrm{m/s} \)
• Acceleration due to gravity \( a = -9.81 \ \mathrm{m/s^2} \)
• Final vertical velocity \( v_y = 0 \), which is the velocity at maximum height

Using the equation \( v_y^2 = v_{0y}^2 + 2ay_{\text{max}} \), we find that the rocket reaches a maximum height of approximately \(81.58 \ \mathrm{m}\). This calculation solely depends on the rocket's vertical launch speed and the constant force of gravity pulling it downward.

Horizontal motion is the motion that happens parallel to the ground. In projectile problems, horizontal motion is often uniform, meaning that it occurs at a constant speed.

For the cart carrying the rocket, its horizontal motion operates independently from the vertical motion of the rocket. The cart moves at \(30.0 \ \mathrm{m/s}\) continuously. As the rocket moves vertically, its horizontal velocity is the same as the cart’s due to it being launched from the cart:

• Constant horizontal speed \( v_x = 30.0 \ \mathrm{m/s} \)
• Horizontal distance traveled while the rocket is in the air \( d = v \times t \)

To calculate the distance covered by the cart during the rocket's flight time \( t = 8.16 \ \mathrm{s} \), we have \( d = 30.0 \times 8.16 = 244.8 \ \mathrm{m}\). This simple multiplication reflects how horizontal movement is frequently uniform, marked by a lack of acceleration, assuming no other forces like air resistance.

Relative velocity examines how the speed and direction of one object is perceived from another object's perspective. In our scenario, understanding that the rocket maintains the same horizontal speed as the cart is central to grasping the idea of relative velocity.

From the perspective of someone on the cart, the missile's vertical launch and eventual landing exhibit no horizontal movement. However, from a stationary viewer on the ground, the missile appears to move horizontally alongside the cart. This stationary bystander sees the rocket traveling the same horizontal distance as the cart:

• Rocket's horizontal speed relative to the cart is \(0 \ \mathrm{m/s}\)
• Rocket's horizontal speed relative to the ground is \(30.0 \ \mathrm{m/s}\)
• Thus, no net horizontal displacement relative to the cart

Hence, relative velocity helps evaluate where the rocket lands. It underscores that from the cart’s perspective, the rocket comes down in the same vertical position from where it was launched.

Gravity is a force that attracts two bodies towards each other, typically pulling objects towards the Earth's center in our day-to-day life. In projectile motion, gravity influences vertical motion by acting as a constant downward force, causing deceleration when moving upwards and acceleration when moving downwards.

In the exercise, gravity is the factor that causes the rocket to decelerate as it rises until it reaches the apex of its trajectory. At this point, vertical velocity is zero. As the rocket descends, gravity accelerates it downward. The key aspects are:

• Gravitational acceleration is \(a = -9.81 \ \mathrm{m/s^2}\)
• Gravity does not affect horizontal motion, thus horizontal motion remains constant
• In the absence of other forces (like significant air resistance), gravity is the sole vertical force

Due to these effects, the total time the rocket spends in the air can be calculated easily through consistent upward and downward travel. Gravity ensures that the rocket's motion is symmetric, where the time taken to ascend equals the time to descend, at \(8.16 \ \mathrm{s}\). This symmetry simplification is common in projectile motion under constant gravity.