Motion in a plane: projectiles and circular motion

A small ball leaves the edge of a table $1.25\mathrm{m}$ high horizontally and hits the floor $1.50\mathrm{m}$ from the vertical through the edge. Neglecting air resistance, compute the velocity with which the ball left the table.

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A projectile is launched from the ground with velocity $30m/s$ inclined at $60^{\circ }$ to the horizontal, on level ground. Neglecting air resistance, compute the time of flight, the maximum height and the range.

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An object launched from the ground must land $50\mathrm{m}$ from the launch point, on level ground. Neglecting air resistance, compute the minimum launch speed that allows this and the corresponding launch angle.

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From a balcony $10\mathrm{m}$ high a stone is thrown with velocity $10m/s$ inclined at $45^{\circ }$ upward. Neglecting air resistance, compute after how much time and at what horizontal distance from the base of the balcony the stone hits the ground, and the magnitude of its velocity at impact.

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A plane flies at $200km/h$ relative to the air and must reach a city located $300\mathrm{km}$ north of the starting point. A uniform wind of $40km/h$ blows toward the east. Compute in which direction the plane must head, its velocity relative to the ground and the duration of the trip.

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A projectile is launched with velocity $15m/s$ inclined at $60^{\circ }$ to the horizontal. Neglecting air resistance, compute the radius of curvature of the trajectory at the highest point.

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A bicycle travels at $18km/h$ and its wheels have radius $0.35\mathrm{m}$. Assuming that the wheels roll without slipping, so that the velocity of the center of the wheel is $\omega R$, compute the angular velocity of the wheels and the number of revolutions per minute.

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A car travels at a constant speed of $72km/h$ around a circular curve of radius $60\mathrm{m}$. Compute its acceleration.

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The International Space Station travels in an orbit that can be considered circular, of radius $6.78\cdot 10^6\mathrm{m}$ measured from the center of the Earth, with a period of $92.7$ minutes. Compute its speed and its centripetal acceleration.

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The shaft of an engine goes from $600$ to $1500$ revolutions per minute in $6.0\mathrm{s}$, with constant angular acceleration. Compute the angular acceleration and the number of revolutions completed in the $6.0\mathrm{s}$.

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A disk starts from rest and rotates with constant angular acceleration $4.0rad/{\mathrm{s}}^2$. For a point $0.20\mathrm{m}$ from the axis, compute tangential acceleration, centripetal acceleration and magnitude of the acceleration at time $t=2.0\mathrm{s}$.

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A point moves with equation of motion $\vec{r}(t)=4.0t\hat{ı}+3.0t^2\hat{ȷ}$, with $\vec{r}$ in meters and $t$ in seconds. At time $t=2.0\mathrm{s}$ compute the magnitude of the velocity, the tangential and normal components of the acceleration and the radius of curvature of the trajectory.

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A swimmer crosses a river $30\mathrm{m}$ wide, swimming perpendicular to the banks with velocity $1.2m/s$ relative to the water; the current has velocity $0.50m/s$. Compute the velocity of the swimmer relative to the bank, the crossing time and the downstream displacement.

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