justagirl
Sep12-04, 11:33 AM
Problem 1
Ball A is dropped from the top of a building of height H at the same instant ball B is thrown vertically upward from the ground. First consider the situation where the balls are moving in opposite directions when the collide. If the speed of ball A is m times the speed of ball B when they collide, find the height at which they collide in terms of H and m. Take x = 0 at the ground, positive upward.
1.1 With two equations, describe the conditions at the collision (position and velocities of the balls).
- I got Xa = H - 1/2gt2
Xb = Vob(t) - 1/2gt2
At collision: Xa = Xb
H = Vob(t)
Va = -gt
Vb = Vob - gt
At collision, Va = Vb
t = mVob/2g
so Va = -g(mVob)/2g
Vb = Vob - g(mVob)/2g
1.2 Write the expressions for position and velocity of the balls as a function of time.
See above
1.3 Solve the above equations to find the height at which the balls collide. Your answer should be expressed as a fraction of the height of the building H and it should depend on the speed ratio m.
I tried solving it but couldn't eliminate the Vob....help!
1.4 Now suppose that m can be negative (i.e. balls A and B are moving in the same direction when they collide). Use the formula derived above to graph the height of the collision (again expressed as a fraction of the building height H) as a function of m for -5 < m < 5. Are there values of m for which the answer is unphysical?
Problem 2
Suppose, for a change, the acceleration of an object is a function of x, where a(x) = bx and b is a constant with a value of 2 seconds-2. In order to solve this problem you should use the chain rule: for arbitrary variables u, v and t, remember that dr/dt = (dr/ds) * (ds/dt).
2.1 If the speed at x = 1 m is zero, what is the speed at x = 3 m? Be sure to show your work.
By integration, should V = x squared?
2.2 How long does it take to travel from x = 1 to x = 3 m?
Problem 3
A small rock sinking through water experiences an exponentially decreasing acceleration as a function of time, given by a(t) = ge-bt, where b is a positive constant that depends on the shape and size of the rock and the physical properties of water.
3.1 Derive an expression for the position of the rock as a function of time, assuming the initial speed of the rock is zero.
3.2 Show that the rock's acceleration can be written in a simple form involving its speed v: a = g - bv (still assuming that its initial speed is zero). This is, perhaps, a more common form of expressing acceleration in the presence of drag.
Ball A is dropped from the top of a building of height H at the same instant ball B is thrown vertically upward from the ground. First consider the situation where the balls are moving in opposite directions when the collide. If the speed of ball A is m times the speed of ball B when they collide, find the height at which they collide in terms of H and m. Take x = 0 at the ground, positive upward.
1.1 With two equations, describe the conditions at the collision (position and velocities of the balls).
- I got Xa = H - 1/2gt2
Xb = Vob(t) - 1/2gt2
At collision: Xa = Xb
H = Vob(t)
Va = -gt
Vb = Vob - gt
At collision, Va = Vb
t = mVob/2g
so Va = -g(mVob)/2g
Vb = Vob - g(mVob)/2g
1.2 Write the expressions for position and velocity of the balls as a function of time.
See above
1.3 Solve the above equations to find the height at which the balls collide. Your answer should be expressed as a fraction of the height of the building H and it should depend on the speed ratio m.
I tried solving it but couldn't eliminate the Vob....help!
1.4 Now suppose that m can be negative (i.e. balls A and B are moving in the same direction when they collide). Use the formula derived above to graph the height of the collision (again expressed as a fraction of the building height H) as a function of m for -5 < m < 5. Are there values of m for which the answer is unphysical?
Problem 2
Suppose, for a change, the acceleration of an object is a function of x, where a(x) = bx and b is a constant with a value of 2 seconds-2. In order to solve this problem you should use the chain rule: for arbitrary variables u, v and t, remember that dr/dt = (dr/ds) * (ds/dt).
2.1 If the speed at x = 1 m is zero, what is the speed at x = 3 m? Be sure to show your work.
By integration, should V = x squared?
2.2 How long does it take to travel from x = 1 to x = 3 m?
Problem 3
A small rock sinking through water experiences an exponentially decreasing acceleration as a function of time, given by a(t) = ge-bt, where b is a positive constant that depends on the shape and size of the rock and the physical properties of water.
3.1 Derive an expression for the position of the rock as a function of time, assuming the initial speed of the rock is zero.
3.2 Show that the rock's acceleration can be written in a simple form involving its speed v: a = g - bv (still assuming that its initial speed is zero). This is, perhaps, a more common form of expressing acceleration in the presence of drag.