Classical Mechanics
# 2D Kinematics

A "drop ball" in tennis is a shot that barely passes the net and touches the other side of the court close to the net. This makes it difficult for the opponent to reach the ball if the opponent starts running for the ball from the baseline.

Suppose you hit a drop ball \(3.6\text{ m}\) from your side of the court, making the ball leave your racket very close to the ground. The ball then clears the net at a height of \(1\text{ m}\) and touches the ground \(0.9\text{ m}\) from the net on the opponent's side, as shown in the diagram.

How long (in seconds) is the ball in the air before hitting the ground after you hit the drop ball, to 2 decimal places?

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**Details and Assumptions:**

- Neglect air resistance.
- The plane containing the trajectory of the ball is perpendicular to the net.
- Use \(g = 10 \text{ ms}^{-2}\).

There is a vertical pole perpendicular to the horizontal plane. From point \(P\) on the plane, 2 projectiles are fired simultaneously at different velocities. The first projectile is fired at an angle of \(30^{\circ}\) and it hits the foot of the pole. The second projectile is fired at an angle of \(60^{\circ}\) and it hits the top of the pole.

It is further known that the projectiles hit the pole at the same time.

If the angle subtended by the pole from \(P\) is \(\alpha\), find \(\tan \alpha\) to 3 decimal places.

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**Details and Assumptions:**

- Air resistance is negligible.
- A gravitational pull is present.

A horizontal plane supports a stationary vertical cylinder of radius \(R\) and a disc \(A\) attached to the cylinder by a horizontal thread \(AB\) of length \(L\) (see figure above, top view). An initial velocity \(v\) is imparted to the disc, as shown above. How long will it move along the plane until it strikes against the cylinder?

**Details and Assumptions:**

- Neglect friction at all contact surfaces.
- Do not consider gravitational forces.
- The radius of the disc as compared to the radius of the cylinder is very small.

A projectile is launched from an initial position \((x,y) = (-\SI{4}{\meter},\SI{0}{\meter})\). The projectile's initial speed is \(12\text{ m/s}\) and it is launched at a \(\SI{75}{\degree}\) angle with respect to the \(x\)-axis. There is a staircase with 4 steps, as pictured above. The top step begins at \((x,y) = (\SI{0}{\meter},\SI{4}{\meter})\) meters and the bottom step ends at \((x,y) = (\SI{4}{\meter},\SI{0}{\meter})\). Each step is \(\SI{1}{\meter}\) wide and \(\SI{1}{\meter}\) deep. The projectile flies over part of the staircase and lands on one of the steps. What is the \(x\) coordinate of the landing point in meters?

**Details and Assumptions**

- The gravitational acceleration is in the \(-y\) direction, with a magnitude of \(\SI{9.8}{\meter\per\second\squared}\).

Tom and Jerry both have equal top running speeds and are initially at points \(A\) and \(B,\) respectively, separated by a distance of \(d\).

They both spot each other and immediately start running at their top speeds. Jerry runs on a straight line perpendicular to the line \(AB\) and Tom runs in such a way that its velocity always points towards the current location of Jerry.

Let \(r(t)\) denote the distance between Tom and Jerry at time \(t\).

Find \(\displaystyle \lim_{t \to \infty} r(t)\).

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