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. A particle moves so that its position vector at time t is given by ˜ r = e −t cost ˜ i + e −t sin t ˜ j. Show that at any time t, (a) its velocity ˜ v is inclined to the vector ˜ r at a constant angle 3π/4 radians. (b) its acceleration vector is at right angles to the vector ˜ r.
2. A particle moves so that its position vector at time t is given by ˜ r = e −t cost ˜ i + e −t sin t ˜ j. Show that at any time t, (a) its velocity ˜ v is inclined to the vector ˜ r at a constant angle 3π/4 radians. (b) its acceleration vector is at right angles to the vector ˜ r.
Two inertial systems are uniformly separating at a speed of exactly √0.84𝑐. In one system a jogger runs a mile (1609m)
in 6 min along the axis of relative motion. How far in meters does he run and how long does it take, to observers in the other
system?
A metal bowl with a weight of 2.25 N is placed in a larger kitchen container filled with soybean oil. How much soybean oil must the bowl displace in order to float? For reference, the mass density of soybean oil is about 917 g/liter and its weight density is about 8.99 N/liter. Please give your answer in liters.
A scale reads 398 N when a piece of copper is hanging from it. What does it read (in N) when it is lowered so that the copper is submerged in water?
Harry Houdini is trapped inside a glass box which weighs 150kg and has dimensions of 0.5m × 0.5m × 2m. The box is tied to an anchor of mass 350kg and is dropped into a lake. In order to escape, he tries to break the glass by kicking it. Will he be able to escape? If so, how much time does he have to break the glass and escape safely?
(1-Dimensional Collision) A 100g solid ball approaches a 1kg solid ball which is also approaching the 100g ball. If the initial speeds of both balls are 5m/s, what are the final speeds of the balls?
(1-Dimensional Collision) A 100g solid ball approaches a 1kg solid ball which is initially at rest. If the initial speed of the 100g ball is 5m/s, what are the final speeds of the balls?
(2-Dimensial Collision) A billiard ball with speed v = 2.0m/s approaches an identical stationary one (see figure at the right). The balls bounce off each other elastically, in such a way that the incoming one gets deflected by an angle θ = 35°. a. What are the final speeds of the balls? b. What is the angle, φ, at which the stationary ball is ejected?
A 15 kg child slides with an initial speed of 1.5 m/s down a playground slide that is 4.0 m long as shown in the figure The slide makes a 40 angle with the horizontal. The child's speed at the bottom is 3.2 m/s i)What was the force of friction that the slide was exerting on the child?
ii) How much work did the friction do?
iii)How much work did gravity do?
iv)How fast would the child travel if friction was negligible ?
v) What is the child's horizontal component of velocity?
