Browse Problems
40 problems tagged with projectile
P0274
Expert Mechanics › KinematicsProduct of Flight Times for Equal-Range Projectiles
Two particles are launched from the same point on the ground with the same initial speed $v_0$ but at different launch angles. Air resistance is negligible.
P0273
Expert Mechanics › KinematicsMaximizing Projectile Range on an Inclined Plane
As shown in Figure, a cannon is situated on a hillside plane with an inclination angle of $\theta$. The cannon fires a projectile with an initial velocity of $v_0$ at an elevation angle of $\alpha$ relative to the slope. Air resistance is to be neglected.
- Find the position where the projectile lands on the slope.
- Find the elevation angle $\alpha$ that achieves the maximum range along the slope.
P0276
Expert Mechanics › KinematicsProjectile Motion of Two Balls Clearing a Rod
As shown in the figure, two balls, Ball 1 and Ball 2, are launched horizontally from the same point at a height $H$ above the ground. The origin O is directly below the launch point. The initial horizontal velocities are $v_1$ and $v_2$ respectively, with $v_1 > v_2$. Ball 1's trajectory just clears the top of a vertical rod of height $h$ located at a horizontal position $x_P$. Ball 1 then lands on the ground at a horizontal distance $R$ from the origin. Ball 2 is launched, lands on the ground at $x=R/3$, undergoes a perfect elastic collision, and its rebound trajectory also just clears the top of the same rod. Ball 2's second landing point is also at $x=R$.
- Find the ratio $v_1/v_2$.
- Find the position of the rod, $x_P$.
- Find the height of the rod, $h$.
P0278
Expert Mechanics › KinematicsProjectile Motion and Parabolic Safety Envelope
A small ball is thrown with an initial velocity $v_0$ in a uniform gravitational field. The plane of motion of the ball is the xz plane. The x-axis is horizontal, and the positive z-axis is opposite to the direction of gravitational acceleration $g$. Air resistance is ignored.
- Assume the ball is launched from the origin with a constant initial speed $v_0$. By adjusting the launch angle, all targets within the region $z \le z_0 - kx^2$ can be hit. This inequality does not need to be proven. Find the expressions for the parameters $z_0$ and $k$.
- Now, the launch point of the ball can be chosen anywhere on the horizontal plane $z=0$, and the launch angle can also be adjusted. The objective is to hit the highest point of a spherical building of radius $R$ (Figure 1.1), which rests on the ground at the origin. The ball must not bounce on the building before hitting the target. Qualitatively determine the shape of the ball's optimal trajectory (the one requiring minimum initial speed).
- In order to hit the highest point of the spherical building of radius $R$, find the expression for the minimum initial launch speed $v_{min}$.
P0305
Intermediate Mechanics › KinematicsProjectile Range from an Elevated Height
A projectile is launched with an initial speed $v_0$ from a height $h$ above the ground. The launch angle is $\theta$ with respect to the horizontal.
P0304
Intermediate Mechanics › KinematicsSymbolic Projectile Motion from a Building
A ball is thrown from the edge of a roof at a height $h$ above the ground. The ball hits the ground after a time $t$ at a horizontal distance $d$ from the building, making an angle $\theta$ with the horizontal.
- Find the height $h$.
- Find the magnitude of the initial velocity, $v_0$.
- Find the angle $\phi_0$ of the initial velocity relative to the horizontal.
- Determine the condition for the initial velocity to be directed above or below the horizontal.
P0290
Intermediate Mechanics › KinematicsProjectile Motion Landing on a Slope
A projectile is launched from the origin with an initial speed $v_0$ at an angle $\theta_0$ above the horizontal. It travels under the influence of gravity $g$. The ground below consists of a flat section followed by a downward slope at an angle $\alpha$ to the horizontal. The angle between the projectile's velocity vector and the slope upon landing is $\phi$.
- In a first case, the projectile lands on the slope at the same vertical level as its launch ($y=0$). Find the angle $\phi$.
- In a second case, the projectile is launched from the edge of the slope and lands on it at a point below the launch level. Find the magnitude of the vertical displacement, $|y|$, for this landing.
- For the landing described in Q2, find the angle $\phi$.
P0294
Intermediate Mechanics › KinematicsProjectile Motion Analysis Over a Wall
A projectile is launched from an initial height $h_0$ and is later caught at the same height. It passes over a wall, first on its way up at time $t_1$ after launch, and again on its way down after an additional time interval $\Delta t$. The horizontal distance between the points where the projectile clears the wall is $D$.
- What is the total horizontal distance $R$ traveled by the projectile?
- What is the magnitude of the projectile's initial velocity, $v_0$?
- What is the angle $\theta_0$ (relative to the horizontal) of the projectile's initial velocity?
- What is the height $H$ of the wall, measured from the ground?
P0295
Intermediate Mechanics › KinematicsProjectile Minimum Speed vs Flight Time
A projectile is launched from level ground with a constant initial speed $v_0$. The launch angle $\theta_0$ can be varied. The maximum possible range is $R_{max}$, and the maximum possible flight time is $t_{max}$.
P0296
Intermediate Mechanics › KinematicsBall Rolling Horizontally Off a Staircase
A ball rolls horizontally off the top of a stairway with an initial speed $v_0$. The steps each have height $h$ and width $w$.
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